Modeling the influence of the E-cadherin-beta-catenin pathway in cancer cell invasion: a multiscale approach

Estimating dose painting effects in radiotherapy: a mathematical model

Tumor heterogeneity is widely considered to be a determinant factor in tumor progression and in particular in its recurrence after therapy. Unfortunately, current medical techniques are unable to deduce clinically relevant information about tumor heterogeneity by means of non-invasive methods. As a consequence, when radiotherapy is used as a treatment of choice, radiation dosimetries are prescribed under the assumption that the malignancy targeted is of a homogeneous nature. In this work we discuss the effects of different radiation dose distributions on heterogeneous tumors by means of an individual cell-based model. To that end, a case is considered where two tumor cell phenotypes are present, which we assume to strongly differ in their respective cell cycle duration and radiosensitivity properties. We show herein that, as a result of such differences, the spatial distribution of the corresponding phenotypes, whence the resulting tumor heterogeneity can be predicted as growth proceeds. In particular, we show that if we start from a situation where a majority of ordinary cancer cells (CCs) and a minority of cancer stem cells (CSCs) are randomly distributed, and we assume that the length of CSC cycle is significantly longer than that of CCs, then CSCs become concentrated at an inner region as tumor grows. As a consequence we obtain that if CSCs are assumed to be more resistant to radiation than CCs, heterogeneous dosimetries can be selected to enhance tumor control by boosting radiation in the region occupied by the more radioresistant tumor cell phenotype. It is also shown that, when compared with homogeneous dose distributions as those being currently delivered in clinical practice, such heterogeneous radiation dosimetries fare always better than their homogeneous counterparts. Finally, limitations to our assumptions and their resulting clinical implications will be discussed.

The effects of cell compressibility, motility and contact inhibition on the growth of tumor cell clusters using the Cellular Potts Model

There are numerous biological examples where genes associated with migratory ability of cells also confer the cells with an increased fitness even though these genes may not have any known effect on the cell mitosis rates. Here, we provide insight into these observations by analyzing the effects of cell migration, compression, and contact inhibition on the growth of tumor cell clusters using the Cellular Potts Model (CPM) in a monolayer geometry. This is a follow-up of a previous study (Thalhauser et al. 2010) in which a Moran-type model was used to study the interaction of cell proliferation, migratory potential and death on the emergence of invasive phenotypes. Here, we extend the study to include the effects of cell size and shape. In particular, we investigate the interplay between cell motility and compressibility within the CPM and find that the CPM predicts that increased cell motility leads to smaller cells. This is an artifact in the CPM. An analysis of the CPM reveals an explicit inverse-relationship between the cell stiffness and motility parameters. We use this relationship to compensate for motility-induced changes in cell size in the CPM so that in the corrected CPM, cell size is independent of the cell motility. We find that subject to comparable levels of compression, clusters of motile cells grow faster than clusters of less motile cells, in qualitative agreement with biological observations and our previous study. Increasing compression tends to reduce growth rates. Contact inhibition penalizes clumped cells by halting their growth and gives motile cells an even greater advantage. Finally, our model predicts cell size distributions that are consistent with those observed in clusters of neuroblastoma cells cultured in low and high density conditions.

Extracellular matrix density promotes EMT by weakening cell–cell adhesions

This paper probes the influence of extracellular matrix density on cell–cell adhesion and its relevance to EMT.

Computational models reveal a passive mechanism for cell migration in the crypt

Cell migration in the intestinal crypt is essential for the regular renewal of the epithelium, and the continued upward movement of cells is a key characteristic of healthy crypt dynamics. However, the driving force behind this migration is unknown. Possibilities include mitotic pressure, active movement driven by motility cues, or negative pressure arising from cell loss at the crypt collar. It is possible that a combination of factors together coordinate migration. Here, three different computational models are used to provide insight into the mechanisms that underpin cell movement in the crypt, by examining the consequence of eliminating cell division on cell movement. Computational simulations agree with existing experimental results, confirming that migration can continue in the absence of mitosis. Importantly, however, simulations allow us to infer mechanisms that are sufficient to generate cell movement, which is not possible through experimental observation alone. The results produced by the three models agree and suggest that cell loss due to apoptosis and extrusion at the crypt collar relieves cell compression below, allowing cells to expand and move upwards. This finding suggests that future experiments should focus on the role of apoptosis and cell extrusion in controlling cell migration in the crypt.

Hybrid model of erythropoiesis

A hybrid model of cell dynamics is presented. It is illustrated by model examples and applied to study erythropoiesis (red blood cell production). In this approach, cells are considered as discrete objects while intra-cellular proteins and extra-cellular biochemical substances are described with continuous models. Spatial organization of erythropoiesis occurring in specific structures of the bone marrow, called erythroblastic island, is investigated.

Toward a quantitative understanding of the Wnt/β-catenin pathway through simulation and experiment

Wnt signaling regulates cell survival, proliferation, and differentiation throughout development and is aberrantly regulated in cancer. The pathway is activated when Wnt ligands bind to specific receptors on the cell surface, resulting in the stabilization and nuclear accumulation of the transcriptional co-activator β-catenin. Mathematical and computational models have been used to study the spatial and temporal regulation of the Wnt/β-catenin pathway and to investigate the functional impact of mutations in key components. Such models range in complexity, from time-dependent, ordinary differential equations that describe the biochemical interactions between key pathway components within a single cell, to complex, multiscale models that incorporate the role of the Wnt/β-catenin pathway target genes in tissue homeostasis and carcinogenesis. This review aims to summarize recent progress in mathematical modeling of the Wnt pathway and to highlight new biological results that could form the basis for future theoretical investigations designed to increase the utility of theoretical models of Wnt signaling in the biomedical arena.

Computational model to probe cellular mechanics during epithelial-mesenchymal transition

During the epithelial to mesenchymal transition (EMT), polarized cells in the epithelium can undergo a transformation characterized by the loss of cell-cell junctions and increased migratory activity into nonpolarized invasive cells. These cells adopt a mesenchymal shape and migrate into the basal lamina. Such transitions have been observed in developmental processes and have been linked to cancer cell metastasis. Most experimental studies on EMT search for molecular markers indicating an epithelial or mesenchymal conformation, focussing on afferent signaling pathways received by cells undergoing this transformation; however, these approaches are unable to track mechanical changes in the cell and the possible role this plays in EMT. In order to address this gap in our understanding, we have used a quantitative approach to study population level effects of single cell changes typically occurring during EMT. We have developed a computational model making use of the advantages of both single cell migratory models and agent-based cell population models to study the effect of cellular molecular processes in EMT. The disruption of a cell sheet representing the epithelium over a dense extracellular matrix (ECM) is simulated using interaction forces between different cells and between cells and discrete fibers representing the ECM. In our study, two different parameters were varied: protrusion force magnitude and E-cadherin (cell-cell junction) concentration. The cell population was tracked for 3 days and the number of cells that leave the layer, the depth of invasion, and the percentage of initial number of cells that remain in the layer (a measure of epithelium disruption) were monitored. Our studies suggest that having a high protrusion force or a reduction in cell-cell attachments is enough to cause EMT. Our results also demonstrate that the morphological progression in membrane disruption has an effect on the number of cells becoming invasive, with epithelial layers broken into clusters hindering the further exodus of cells. The results reveal the quantitative interplay between two key parameters involved in EMT and suggest potential avenues for further exploration of a systems level understanding of EMT.

Modeling the mechanics of cancer: effect of changes in cellular and extra-cellular mechanical properties

Malignant transformation, though primarily driven by genetic mutations in cells, is also accompanied by specific changes in cellular and extra-cellular mechanical properties such as stiffness and adhesivity. As the transformed cells grow into tumors, they interact with their surroundings via physical contacts and the application of forces. These forces can lead to changes in the mechanical regulation of cell fate based on the mechanical properties of the cells and their surrounding environment. A comprehensive understanding of cancer progression requires the study of how specific changes in mechanical properties influences collective cell behavior during tumor growth and metastasis. Here we review some key results from computational models describing the effect of changes in cellular and extra-cellular mechanical properties and identify mechanistic pathways for cancer progression that can be targeted for the prediction, treatment, and prevention of cancer.

Dealing with diversity in computational cancer modeling

This paper discusses the need for interconnecting computational cancer models from different sources and scales within clinically relevant scenarios to increase the accuracy of the models and speed up their clinical adaptation, validation, and eventual translation. We briefly review current interoperability efforts drawing upon our experiences with the development of in silico models for predictive oncology within a number of European Commission Virtual Physiological Human initiative projects on cancer. A clinically relevant scenario, addressing brain tumor modeling that illustrates the need for coupling models from different sources and levels of complexity, is described. General approaches to enabling interoperability using XML-based markup languages for biological modeling are reviewed, concluding with a discussion on efforts towards developing cancer-specific XML markup to couple multiple component models for predictive in silico oncology.

In vivo mathematical modeling of tumor growth from imaging data: soon to come in the future?

The future challenges in oncology imaging are to assess the response to treatment even earlier. As an addition to functional imaging, mathematical modeling based on the imaging is an alternative, cross-disciplinary area of development. Modeling was developed in oncology not only in order to understand and predict tumor growth, but also to anticipate the effects of targeted and untargeted therapies. A very wide range of these models exist, involving many stages in the progression of tumors. Few models, however, have been proposed to reproduce in vivo tumor growth because of the complexity of the mechanisms involved. Morphological imaging combined with "spatial" models appears to perform well although functioning imaging could still provide further information on metabolism and the micro-architecture. The combination of imaging and modeling can resolve complex problems and describe many facets of tumor growth or response to treatment. It is now possible to consider its clinical use in the medium term. This review describes the basic principles of mathematical modeling and describes the advantages, limitations and future prospects for this in vivo approach based on imaging data.

Dynamic transcription factor networks in epithelial-mesenchymal transition in breast cancer models

The epithelial-mesenchymal transition (EMT) is a complex change in cell differentiation that allows breast carcinoma cells to acquire invasive properties. EMT involves a cascade of regulatory changes that destabilize the epithelial phenotype and allow mesenchymal features to manifest. As transcription factors (TFs) are upstream effectors of the genome-wide expression changes that result in phenotypic change, understanding the sequential changes in TF activity during EMT provides rich information on the mechanism of this process. Because molecular interactions will vary as cells progress from an epithelial to a mesenchymal differentiation program, dynamic networks are needed to capture the changing context of molecular processes. In this study we applied an emerging high-throughput, dynamic TF activity array to define TF activity network changes in three cell-based models of EMT in breast cancer based on HMLE Twist ER and MCF-7 mammary epithelial cells. The TF array distinguished conserved from model-specific TF activity changes in the three models. Time-dependent data was used to identify pairs of TF activities with significant positive or negative correlation, indicative of interdependent TF activity throughout the six-day study period. Dynamic TF activity patterns were clustered into groups of TFs that change along a time course of gene expression changes and acquisition of invasive capacity. Time-dependent TF activity data was combined with prior knowledge of TF interactions to construct dynamic models of TF activity networks as epithelial cells acquire invasive characteristics. These analyses show EMT from a unique and targetable vantage and may ultimately contribute to diagnosis and therapy.

A computational model for collective cellular motion in three dimensions: general framework and case study for cell pair dynamics

Cell migration in healthy and diseased systems is a combination of single and collective cell motion. While single cell motion has received considerable attention, our understanding of collective cell motion remains elusive. A new computational framework for the migration of groups of cells in three dimensions is presented, which focuses on the forces acting at the microscopic scale and the interactions between cells and their extracellular matrix (ECM) environment. Cell-cell adhesion, resistance due to the ECM and the factors regulating the propulsion of each cell through the matrix are considered. In particular, our approach emphasizes the role of receptors that mediate cell-cell and cell-matrix interactions, and examines how variation in their properties induces changes in cellular motion. As an important case study, we analyze two interacting cells. Our results show that the dynamics of cell pairs depends on the magnitude and the stochastic nature of the forces. Stronger intercellular stability is generally promoted by surface receptors that move. We also demonstrate that matrix resistance, cellular stiffness and intensity of adhesion contribute to migration behaviors in different ways, with memory effects present that can alter pair motility. If adhesion weakens with time, our findings show that cell pair break-up depends strongly on the way cells interact with the matrix. Finally, the motility for cells in a larger cluster (size 50 cells) is examined to illustrate the full capabilities of the model and to stress the role of cellular pairs in complex cellular structures. Overall, our framework shows how properties of cells and their environment influence the stability and motility of cellular assemblies. This is an important step in the advancement of the understanding of collective motility, and can contribute to knowledge of complex biological processes involving migration, aggregation and detachment of cells in healthy and diseased systems.

Computational modeling of single-cell migration: the leading role of extracellular matrix fibers

Cell migration is vitally important in a wide variety of biological contexts ranging from embryonic development and wound healing to malignant diseases such as cancer. It is a very complex process that is controlled by intracellular signaling pathways as well as the cell's microenvironment. Due to its importance and complexity, it has been studied for many years in the biomedical sciences, and in the last 30 years it also received an increasing amount of interest from theoretical scientists and mathematical modelers. Here we propose a force-based, individual-based modeling framework that links single-cell migration with matrix fibers and cell-matrix interactions through contact guidance and matrix remodelling. With this approach, we can highlight the effect of the cell's environment on its migration. We investigate the influence of matrix stiffness, matrix architecture, and cell speed on migration using quantitative measures that allow us to compare the results to experiments.

Expression of glioma-associated oncogene 2 (Gli 2) is correlated with poor prognosis in patients with hepatocellular carcinoma undergoing hepatectomy

Background

Our previous studies showed that glioma-associated oncogene (Gli)2 plays an important role in the proliferation and apoptosis resistance of hepatocellular carcinoma (HCC) cells. The aim of this study was to explore the clinical significance of Gli2 expression in HCC.

Methods

Expression of Gli2 protein was detected in samples from 68 paired HCC samples, the corresponding paraneoplastic liver tissues, and 20 normal liver tissues using immunohistochemistry. Correlation of the immunohistochemistry results with clinicopathologic parameters, prognosis, and the expression of E-cadherin, N-cadherin, and vimentin were analyzed.

Results

Immunohistochemical staining showed high levels of Gli2 protein expression in HCC, compared with paraneoplastic and normal liver tissues (P < 0.05). This high expression level of Gli2 was significantly associated with tumor differentiation, encapsulation, vascular invasion, early recurrence, and intra-hepatic metastasis (P < 0.05). There was a significantly negative correlation between Gli2 and E-cadherin expression (r = −0.302, P < 0.05) and a significantly positive correlation between expression of Gli2 and expression of vimentin (r = −0.468, P < 0.05) and N-cadherin (r = −0.505, P < 0.05). Kaplan-Meier analysis showed that patients with overexpressed Gli2 had significantly shorter overall survival and disease-free survival times (P < 0.05). Multivariate analysis suggested that the level of Gli2 expression was an independent prognostic factor for HCC.

Conclusions

Expression of Gli2 is high in HCC tissue, and is associated with poor prognosis in patients with HCC after hepatectomy.

The tumor growth paradox and immune system-mediated selection for cancer stem cells

Cancer stem cells (CSCs) drive tumor progression, metastases, treatment resistance, and recurrence. Understanding CSC kinetics and interaction with their nonstem counterparts (called tumor cells, TCs) is still sparse, and theoretical models may help elucidate their role in cancer progression. Here, we develop a mathematical model of a heterogeneous population of CSCs and TCs to investigate the proposed "tumor growth paradox"--accelerated tumor growth with increased cell death as, for example, can result from the immune response or from cytotoxic treatments. We show that if TCs compete with CSCs for space and resources they can prevent CSC division and drive tumors into dormancy. Conversely, if this competition is reduced by death of TCs, the result is a liberation of CSCs and their renewed proliferation, which ultimately results in larger tumor growth. Here, we present an analytical proof for this tumor growth paradox. We show how numerical results from the model also further our understanding of how the fraction of cancer stem cells in a solid tumor evolves. Using the immune system as an example, we show that induction of cell death can lead to selection of cancer stem cells from a minor subpopulation to become the dominant and asymptotically the entire cell type in tumors.

The modulatory effect of cell–cell contact on the tumourigenic potential of pre-malignant epithelial cells: a computational exploration

Malignant development cannot be attributed alone to genetic changes in a single cell, but occurs as a result of the complex interplay between the failure of cellular regulation mechanisms and the presence of a permissive microenvironment. Although E-cadherin is classified as a 'metastasis suppressor' owing to its role in intercellular adhesion, the observation that it may be downregulated at a premalignant stage is indicative of additional roles in neoplastic development. We have used an agent-based computational model to explore the emergent behaviour resulting from the interaction of single and subpopulations of E-cadherin-compromised cells with unaffected normal epithelial cells within a monolayer environment. We have extended this to investigate the importance of local tissue perturbations in the form of scratch-wounding, or ablation of randomly-dispersed normal cells, on the growth of a single cell exhibiting E-cadherin loss. Our results suggest that the microenvironment with respect to localized cell density and normal/E-cadherin-compromised neighbours is crucial in determining whether an abnormal individual cell proliferates or remains dormant within the monolayer. These predictions raise important questions relating to the propensity for individual mutations to give rise to disease, and future experimental exploration of these will enhance our understanding of a complex, multifactorial pathological process.

Computational modelling and analysis of mechanical conditions on cell locomotion and cell-cell interaction

Between other parameters, cell migration is partially guided by the mechanical properties of its substrate. Although many experimental works have been developed to understand the effect of substrate mechanical properties on cell migration, accurate 3D cell locomotion models have not been presented yet. In this paper, we present a novel 3D model for cells migration. In the presented model, we assume that a cell follows two main processes: in the first process, it senses its interface with the substrate to determine the migration direction and in the second process, it exerts subsequent forces to move. In the presented model, cell traction forces are considered to depend on cell internal deformation during the sensing step. A random protrusion force is also considered which may change cell migration direction and/or speed. The presented model was applied for many cases of migration of the cells. The obtained results show high agreement with the available experimental and numerical data.

Advances in the formation, use and understanding of multi-cellular spheroids

INTRODUCTION

Developing in vitro models for studying cell biology and cell physiology is of great importance to the fields of biotechnology, cancer research, drug discovery, toxicity testing, as well as the emerging fields of tissue engineering and regenerative medicine. Traditional two-dimensional (2D) methods of mammalian cell culture have several limitations and it is increasingly recognized that cells grown in a three-dimensional (3D) environment more closely represent normal cellular function due to the increased cell-to-cell interactions, and by mimicking the in vivo architecture of natural organs and tissues.

AREAS COVERED

In this review, we discuss the methods to form 3D multi-cellular spheroids, the advantages and limitations of these methods, and assays used to characterize the function of spheroids. The use of spheroids has led to many advances in basic cell sciences, including understanding cancer cell interactions, creating models for drug discovery and cancer metastasis, and they are being investigated as basic units for engineering tissue constructs. As so, this review will focus on contributions made to each of these fields using spheroid models.

EXPERT OPINION

Multi-cellular spheroids are rich in biological content and mimic better the in vivo environment than 2D cell culture. New technologies to form and analyze spheroids are rapidly increasing their adoption and expanding their applications.

The contribution of age structure to cell population responses to targeted therapeutics

Cells grown in culture act as a model system for analyzing the effects of anticancer compounds, which may affect cell behavior in a cell cycle position-dependent manner. Cell synchronization techniques have been generally employed to minimize the variation in cell cycle position. However, synchronization techniques are cumbersome and imprecise and the agents used to synchronize the cells potentially have other unknown effects on the cells. An alternative approach is to determine the age structure in the population and account for the cell cycle positional effects post hoc. Here we provide a formalism to use quantifiable lifespans from live cell microscopy experiments to parameterize an age-structured model of cell population response.

Quantitative approaches to cancer stem cells and epithelial-mesenchymal transition

The last decade has witnessed significant advances in the application of mathematical and computational models to biological systems, especially to cancer biology. Here, we present stochastic and deterministic models describing tumour growth based on the cancer stem cell hypothesis, and discuss the application of these models to the epithelial-mesenchymal transition. In particular, we discuss how such quantitative approaches can be used to validate different possible scenarios that can lead to an increase in stem cell activity following induction of epithelial-mesenchymal transition, observed in recent experimental studies on human breast cancer and related cell lines. The utility of comparing mammosphere data to computational mammosphere simulations in elucidating the growth characteristics of mammary (cancer) stem cells is discussed as well.

Integrating intracellular dynamics using CompuCell3D and Bionetsolver: applications to multiscale modelling of cancer cell growth and invasion

In this paper we present a multiscale, individual-based simulation environment that integrates CompuCell3D for lattice-based modelling on the cellular level and Bionetsolver for intracellular modelling. CompuCell3D or CC3D provides an implementation of the lattice-based Cellular Potts Model or CPM (also known as the Glazier-Graner-Hogeweg or GGH model) and a Monte Carlo method based on the metropolis algorithm for system evolution. The integration of CC3D for cellular systems with Bionetsolver for subcellular systems enables us to develop a multiscale mathematical model and to study the evolution of cell behaviour due to the dynamics inside of the cells, capturing aspects of cell behaviour and interaction that is not possible using continuum approaches. We then apply this multiscale modelling technique to a model of cancer growth and invasion, based on a previously published model of Ramis-Conde et al. (2008) where individual cell behaviour is driven by a molecular network describing the dynamics of E-cadherin and β-catenin. In this model, which we refer to as the centre-based model, an alternative individual-based modelling technique was used, namely, a lattice-free approach. In many respects, the GGH or CPM methodology and the approach of the centre-based model have the same overall goal, that is to mimic behaviours and interactions of biological cells. Although the mathematical foundations and computational implementations of the two approaches are very different, the results of the presented simulations are compatible with each other, suggesting that by using individual-based approaches we can formulate a natural way of describing complex multi-cell, multiscale models. The ability to easily reproduce results of one modelling approach using an alternative approach is also essential from a model cross-validation standpoint and also helps to identify any modelling artefacts specific to a given computational approach.

Multiscale cancer modeling

Simulating cancer behavior across multiple biological scales in space and time, i.e., multiscale cancer modeling, is increasingly being recognized as a powerful tool to refine hypotheses, focus experiments, and enable more accurate predictions. A growing number of examples illustrate the value of this approach in providing quantitative insights in the initiation, progression, and treatment of cancer. In this review, we introduce the most recent and important multiscale cancer modeling works that have successfully established a mechanistic link between different biological scales. Biophysical, biochemical, and biomechanical factors are considered in these models. We also discuss innovative, cutting-edge modeling methods that are moving predictive multiscale cancer modeling toward clinical application. Furthermore, because the development of multiscale cancer models requires a new level of collaboration among scientists from a variety of fields such as biology, medicine, physics, mathematics, engineering, and computer science, an innovative Web-based infrastructure is needed to support this growing community.

Dynamic density functional theory of solid tumor growth: Preliminary models

Cancer is a disease that can be seen as a complex system whose dynamics and growth result from nonlinear processes coupled across wide ranges of spatio-temporal scales. The current mathematical modeling literature addresses issues at various scales but the development of theoretical methodologies capable of bridging gaps across scales needs further study. We present a new theoretical framework based on Dynamic Density Functional Theory (DDFT) extended, for the first time, to the dynamics of living tissues by accounting for cell density correlations, different cell types, phenotypes and cell birth/death processes, in order to provide a biophysically consistent description of processes across the scales. We present an application of this approach to tumor growth.

Patient-calibrated agent-based modelling of ductal carcinoma in situ (DCIS): from microscopic measurements to macroscopic predictions of clinical progression

Ductal carcinoma in situ (DCIS)--a significant precursor to invasive breast cancer--is typically diagnosed as microcalcifications in mammograms. However, the effective use of mammograms and other patient data to plan treatment has been restricted by our limited understanding of DCIS growth and calcification. We develop a mechanistic, agent-based cell model and apply it to DCIS. Cell motion is determined by a balance of biomechanical forces. We use potential functions to model interactions with the basement membrane and amongst cells of unequal size and phenotype. Each cell's phenotype is determined by genomic/proteomic- and microenvironment-dependent stochastic processes. Detailed "sub-models" describe cell volume changes during proliferation and necrosis; we are the first to account for cell calcification. We introduce the first patient-specific calibration method to fully constrain the model based upon clinically-accessible histopathology data. After simulating 45 days of solid-type DCIS with comedonecrosis, the model predicts: necrotic cell lysis acts as a biomechanical stress relief and is responsible for the linear DCIS growth observed in mammography; the rate of DCIS advance varies with the duct radius; the tumour grows 7-10mm per year--consistent with mammographic data; and the mammographic and (post-operative) pathologic sizes are linearly correlated--in quantitative agreement with the clinical literature. Patient histopathology matches the predicted DCIS microstructure: an outer proliferative rim surrounds a stratified necrotic core with nuclear debris on its outer edge and calcification in the centre. This work illustrates that computational modelling can provide new insight on the biophysical underpinnings of cancer. It may 1-day be possible to augment a patient's mammography and other imaging with rigorously-calibrated models that help select optimal surgical margins based upon the patient's histopathologic data.

Invasion from a cell aggregate--the roles of active cell motion and mechanical equilibrium

Cell invasion from an aggregate into a surrounding extracellular matrix (ECM) is an important process during development disease, e.g., vascular network assembly or tumor progression. To describe the behavior emerging from autonomous cell motility, cell-cell adhesion and contact guidance by ECM filaments, we propose a suitably modified cellular Potts model. We consider an active cell motility process in which internal polarity is governed by a positive feedback from cell displacements, a mechanism that can result in highly persistent motion when constrained by an oriented ECM structure. The model allows us to explore the interplay between haptotaxis, matrix degradation and active cell movement. We show that for certain conditions the cells are able to both invade the ECM and follow the ECM tracks. Furthermore, we argue that enforcing mechanical equilibrium within a bulk cell mass is of key importance in multicellular simulations.

Analysis of cell adhesion during early stages of colon cancer based on an extended multi-valued logic approach

Cell adhesion in the normal colon is typically associated with differentiated cells, whereas in cancerous colon it is associated with advanced tumors. For advanced tumors growing evidence supports the existence of stem-like cells that have originated from transdifferentiation. Because stem cells can also be transformed in their own niche, at the base of the Lieberkühn's crypts, we conjectured that cell adhesion can also be critical in early tumorigenesis. To assess this hypothesis we built an annotated, multi-valued logic model addressing cell adhesion of normal and tumorigenic stem cells in the human colon. The model accounts for (i) events involving intercellular adhesion structures, (ii) interactions involving cytoskeleton-related structures, (iii) compartmental distribution of α/β/γ/δ-catenins, and (iv) variations in critical cell adhesion regulators (e.g., ILK, FAK, IQGAP, SNAIL, Caveolin). We developed a method that can deal with graded multiple inhibitions, something which is not possible with conventional logical approaches. The model comprises 315 species (including 26 genes), interconnected by 269 reactions. Simulations of the model covered six scenarios, which considered two types of colonic cells (stem vs. differentiated cells), under three conditions (normal, stressed and tumor). Each condition results from the combination of 92 inputs. We compared our multi-valued logic approach with the conventional Boolean approach for one specific example and validated the predictions against published data. Our analysis suggests that stem cells in their niche synthesize high levels of cytoplasmatic E-cadherin and CdhEP(Ser684,686,692), even under normal-mitogenic stimulus or tumorigenic conditions. Under these conditions, E-cadherin would be incorporated into the plasmatic membrane, but only as a non-adhesive CdhE_β-catenin_IQGAP complex. Under stress conditions, however, this complex could be displaced, yielding adhesive CdhE_β-catenin((cis/trans)) complexes. In the three scenarios tested with stem cells, desmosomes or tight junctions were not assembled. Other model predictions include expected levels of the nuclear complex β-catenin_TCF4 and the anti-apoptotic protein Survivin for both normal and tumorigenic colonic stem cells.

The virtual liver: a multidisciplinary, multilevel challenge for systems biology

The liver is the central metabolic organ in human physiology, with functions that are fundamentally important to the detoxification of xenobiotics (drugs), the maintenance of homeostasis of numerous blood metabolites, and the production of mediators of the acute phase response. Liver toxicity, whether actual or implied is the reason for the failure of a significant proportion of many promising novel medicines that consequently never reach the market, and diseases such as atherosclerosis, diabetes, and fatty liver diseases, that are a major burden on current health resources, are directly linked to functional and structural disorders of the liver. This article presents the concepts and approaches underpinning one of the most exciting and ambitious modeling projects in the field of systems biology and systems medicine. This major multidisciplinary research program is aimed at developing a whole-organ model of the human liver, representing its central physiological functions under normal and pathological conditions The model will be composed of a larger battery of interconnected submodels representing liver anatomy and physiology, integrating processes across hierarchical levels in space, time, and structural organization. In this article, we outline the general architecture of the liver model and present first step taken to reach this ambitious goal.

Normal morphogenesis of epithelial tissues and progression of epithelial tumors

Epithelial cells organize into various tissue architectures that largely maintain their structure throughout the life of an organism. For decades, the morphogenesis of epithelial tissues has fascinated scientists at the interface of cell, developmental, and molecular biology. Systems biology offers ways to combine knowledge from these disciplines by building integrative models that are quantitative and predictive. Can such models be useful for gaining a deeper understanding of epithelial morphogenesis? Here, we take inventory of some recurring themes in epithelial morphogenesis that systems approaches could strive to capture. Predictive understanding of morphogenesis at the systems level would prove especially valuable for diseases such as cancer, where epithelial tissue architecture is profoundly disrupted.

Cancer cell detection in tissue sections using AFM

Currently, cancer diagnosis relies mostly on morphological examination of exfoliated, aspirated cells or surgically removed tissue. As long as standard diagnosis is concerned, this classical approach seems to be satisfactory. In the recent years, cancer progression has been shown to be accompanied by alterations in mechanical properties of cells. This offers the detection of otherwise unnoticed cancer cell disregarded by histological analysis due to insignificant manifestations. One of techniques, sensitive to changes in mechanical properties, is the atomic force microscopy, which detects cancer cells through their elastic properties. Such measurements were applied to tissue sections collected from patients suffering from various cancers. Despite of heterogeneity and complexity of cancer cell sections, the use of the Young's modulus as an indicator of cell elasticity allow for detection of cancer cells in tissue slices.

Differential roles of STAT3 depending on the mechanism of STAT3 activation in gastric cancer cells

Background:

Signal transducer and activator of transcription 3 (STAT3) is a transcription factor that is activated in response to growth factors and cytokines, and which contributes to the regulation of cell proliferation, apoptosis, and motility in many human tumour types.

Methods:

We investigated the mechanisms of STAT3 activation and the function of STAT3 depending on its mechanism of activation in gastric cancer cells.

Results:

The MET-tyrosine kinase inhibitor (TKI) and cell transfection with a small interfering RNA (siRNA) specific for MET mRNA inhibited STAT3 phosphorylation in MET-activated cells, indicating that STAT3 activation is linked to MET signalling. Forced expression of a constitutively active form of STAT3 also attenuated MET-TKI-induced apoptosis, suggesting that inhibition of STAT3 activity contributes to MET-TKI-induced apoptosis. MKN1 and MKN7 cells, both of which are negative for MET activation, produced interleukin-6 (IL-6) that activated STAT3 through the Janus kinase pathway. Depletion of STAT3 by siRNA inhibited migration and invasion of these cells, suggesting that STAT3 activated by IL-6 contributes to regulation of cell motility.

Conclusion:

Our data thus show that activated STAT3 contributes to either cell survival or motility in gastric cancer cells, and that these actions are related to different mechanisms of STAT3 activation.

Hybrid models of tumor growth

Cancer is a complex, multiscale process in which genetic mutations occurring at a subcellular level manifest themselves as functional changes at the cellular and tissue scale. The multiscale nature of cancer requires mathematical modeling approaches that can handle multiple intracellular and extracellular factors acting on different time and space scales. Hybrid models provide a way to integrate both discrete and continuous variables that are used to represent individual cells and concentration or density fields, respectively. Each discrete cell can also be equipped with submodels that drive cell behavior in response to microenvironmental cues. Moreover, the individual cells can interact with one another to form and act as an integrated tissue. Hybrid models form part of a larger class of individual-based models that can naturally connect with tumor cell biology and allow for the integration of multiple interacting variables both intrinsically and extrinsically and are therefore perfectly suited to a systems biology approach to tumor growth.

An Adaptive Multigrid Algorithm for Simulating Solid Tumor Growth Using Mixture Models

In this paper we give the details of the numerical solution of a three-dimensional multispecies diffuse interface model of tumor growth, which was derived in (Wise et al., J. Theor. Biol. 253 (2008)) and used to study the development of glioma in (Frieboes et al., NeuroImage 37 (2007) and tumor invasion in (Bearer et al., Cancer Research, 69 (2009)) and (Frieboes et al., J. Theor. Biol. 264 (2010)). The model has a thermodynamic basis, is related to recently developed mixture models, and is capable of providing a detailed description of tumor progression. It utilizes a diffuse interface approach, whereby sharp tumor boundaries are replaced by narrow transition layers that arise due to differential adhesive forces among the cell-species. The model 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. Numerical solution of the model is challenging because the equations are coupled, highly nonlinear, and numerically stiff. In this paper we describe a fully adaptive, nonlinear multigrid/finite difference method for efficiently solving the equations. We demonstrate the convergence of the algorithm and we present simulations of tumor growth in 2D and 3D that demonstrate the capabilities of the algorithm in accurately and efficiently simulating the progression of tumors with complex morphologies.

Discovering Molecular Targets in Cancer with Multiscale Modeling

Multiscale modeling is increasingly being recognized as a promising research area in computational cancer systems biology. Here, exemplified by two pioneering studies, we attempt to explain why and how such a multiscale approach paired with an innovative cross-scale analytical technique can be useful in identifying high-value molecular therapeutic targets. This novel, integrated approach has the potential to offer a more effective in silico framework for target discovery and represents an important technical step towards systems medicine.

Computational investigation of intrinsic and extrinsic mechanisms underlying the formation of carcinoma

The disruption of tissue epithelial architecture is thought to be involved in the initiation of cancer; however, little is known about the cell biology of early neoplastic lesions. Computational models can facilitate in the study of early cancer stagesby simulating different scenarios of tumour inception and by testing in silico the influence of various intrinsic and extrinsic factors on the emerging multicellular morphologies. Here, we have used a computational model to create 3D morphocharts, collections of multicellular morphologies arising from systematic modification of three model parameters defining cell sensitivity to external cues for cell growth, death and adhesion to the extracellular matrix. This systematic search revealed the ranges of parameter values for which robust epithelial structures formed and those that led to abnormal geometrical forms resembling tumour-like morphologies. We also showed how our model can be used to map morphologies of experimental multicellular systems onto their in silico counterpart via the cell sensitivity parameter space that may in turn allow us to identify genetic or epigenetic changes responsible for these morphological distortions.

Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation

The ability of cancer cells to break out of tissue compartments and invade locally gives solid tumours a defining deadly characteristic. One of the first steps of invasion is the remodelling of the surrounding tissue or extracellular matrix (ECM) and a major part of this process is the over-expression of proteolytic enzymes, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs), by the cancer cells to break down ECM proteins. Degradation of the matrix enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body, a process known as metastasis. In this paper we undertake an analysis of a mathematical model of cancer cell invasion of tissue, or ECM, which focuses on the role of the urokinase plasminogen activation system. The model consists of a system of five reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, uPA, uPA inhibitors, plasmin and the host tissue. Cancer cells react chemotactically and haptotactically to the spatio-temporal effects of the uPA system. The results obtained from computational simulations carried out on the model equations produce dynamic heterogeneous spatio-temporal solutions and using linear stability analysis we show that this is caused by a taxis-driven instability of a spatially homogeneous steady-state. Finally we consider the biological implications of the model results, draw parallels with clinical samples and laboratory based models of cancer cell invasion using three-dimensional invasion assay, and go on to discuss future development of the model.

Mathematical modelling of Wnt/β-catenin signalling

The Wnt/β-catenin pathway plays an important role in development and disease. Theoretical approaches have been used to describe this pathway and have provided intriguing insights into its signalling characteristics. In the present paper, we review mathematical models of the pathway. We focus on a quantitative kinetic model for canonical Wnt/β-catenin signalling describing the reactions of the pathway's core compounds [Lee, Salic, Krüger, Heinrich and Kirschner (2003) PLoS Biol. 1, 116–132]. Numerous modifications and further analyses with respect to signalling characteristics, transcriptional feedback and cross-talk were performed. In addition, the role of β-catenin in gene expression and cell–cell adhesion as well as spatial aspects of signalling are investigated in various theoretical models.

Mathematical Oncology: How Are the Mathematical and Physical Sciences Contributing to the War on Breast Cancer?

Mathematical modeling has recently been added as a tool in the fight against cancer. The field of mathematical oncology has received great attention and increased enormously, but over-optimistic estimations about its ability have created unrealistic expectations. We present a critical appraisal of the current state of mathematical models of cancer. Although the field is still expanding and useful clinical applications may occur in the future, managing over-expectation requires the proposal of alternative directions for mathematical modeling. Here, we propose two main avenues for this modeling: 1) the identification of the elementary biophysical laws of cancer development, and 2) the development of a multiscale mathematical theory as the framework for models predictive of tumor growth. Finally, we suggest how these new directions could contribute to addressing the current challenges of understanding breast cancer growth and metastasis.

A cell-based simulation software for multi-cellular systems

CellSys is a modular software tool for efficient off-lattice simulation of growth and organization processes in multi-cellular systems in 2D and 3D. It implements an agent-based model that approximates cells as isotropic, elastic and adhesive objects. Cell migration is modeled by an equation of motion for each cell. The software includes many modules specifically tailored to support the simulation and analysis of virtual tissues including real-time 3D visualization and VRML 2.0 support. All cell and environment parameters can be independently varied which facilitates species specific simulations and allows for detailed analyses of growth dynamics and links between cellular and multi-cellular phenotypes.

Availability: CellSys is freely available for non-commercial use at http://msysbio.com/software/cellsys. The current version of CellSys permits the simulation of growing monolayer cultures and avascular tumor spheroids in liquid environment. Further functionality will be made available ongoing with published papers.

Contact:hoehme@izbi.uni-leipzig.de; dirk.drasdo@inria.fr

Supplementary information:Supplementary data are available at Bioinformatics online.

Development of a three dimensional multiscale computational model of the human epidermis

Transforming Growth Factor (TGF-β1) is a member of the TGF-beta superfamily ligand-receptor network. and plays a crucial role in tissue regeneration. The extensive in vitro and in vivo experimental literature describing its actions nevertheless describe an apparent paradox in that during re-epithelialisation it acts as proliferation inhibitor for keratinocytes. The majority of biological models focus on certain aspects of TGF-β1 behaviour and no one model provides a comprehensive story of this regulatory factor's action. Accordingly our aim was to develop a computational model to act as a complementary approach to improve our understanding of TGF-β1. In our previous study, an agent-based model of keratinocyte colony formation in 2D culture was developed. In this study this model was extensively developed into a three dimensional multiscale model of the human epidermis which is comprised of three interacting and integrated layers: (1) an agent-based model which captures the biological rules governing the cells in the human epidermis at the cellular level and includes the rules for injury induced emergent behaviours, (2) a COmplex PAthway SImulator (COPASI) model which simulates the expression and signalling of TGF-β1 at the sub-cellular level and (3) a mechanical layer embodied by a numerical physical solver responsible for resolving the forces exerted between cells at the multi-cellular level. The integrated model was initially validated by using it to grow a piece of virtual epidermis in 3D and comparing the in virtuo simulations of keratinocyte behaviour and of TGF-β1 signalling with the extensive research literature describing this key regulatory protein. This research reinforces the idea that computational modelling can be an effective additional tool to aid our understanding of complex systems. In the accompanying paper the model is used to explore hypotheses of the functions of TGF-β1 at the cellular and subcellular level on different keratinocyte populations during epidermal wound healing.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

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.

Computational modeling of epithelial-mesenchymal transformations

An epithelial-mesenchymal transformation (EMT) involves alterations in cell-cell and cell-matrix adhesion, the detachment of epithelial cells from their neighbors, the degradation of the basal lamina and acquisition of mesenchymal phenotype. Here we present Monte Carlo simulations for a specific EMT in early heart development: the formation of cardiac cushions. Cell rearrangements are described in accordance with Steinberg's differential adhesion hypothesis, which states that cells possess a type-dependent adhesion apparatus and are sufficiently motile to give rise to the tissue conformation with the largest number of strong bonds. We also implement epithelial and mesenchymal cell proliferation, cell type change and extracellular matrix production by mesenchymal cells. Our results show that an EMT is promoted more efficiently by an increase in cell-substrate adhesion than by a decrease in cell-cell adhesion. In addition to cushion tissue formation, the model also accounts for the phenomena of matrix invasion and mesenchymal condensation. We conclude that in order to maintain epithelial integrity during EMT the number of epithelial cells must increase at a controlled rate. Our model predictions are in qualitative agreement with available experimental data.

The cellular basis of cell sorting kinetics

Cell sorting is a dynamical cooperative phenomenon that is fundamental for tissue morphogenesis and tissue homeostasis. According to Steinberg's differential adhesion hypothesis, the structure of sorted cell aggregates is determined by physical characteristics of the respective tissues, the tissue surface tensions. Steinberg postulated that tissue surface tensions result from quantitative differences in intercellular adhesion. Several experiments in cell cultures as well as in developing organisms support this hypothesis. The question of how tissue surface tension might result from differential adhesion was addressed in some theoretical models. These models describe the cellular interdependence structure once the temporal evolution has stabilized. In general, these models are capable of reproducing sorted patterns. However, the model dynamics at the cellular scale are defined implicitly and are not well-justified. The precise mechanism describing how differential adhesion generates the observed sorting kinetics at the tissue level is still unclear. It is necessary to formulate the concepts of cell level kinetics explicitly. Only then it is possible to understand the temporal development at the cellular and tissue scales. Here we argue that individual cell mobility is reduced the more the cells stick to their neighbors. We translate this assumption into a precise mathematical model which belongs to the class of stochastic interacting particle systems. Analyzing this model, we are able to predict the emergent sorting behavior at the population level. We describe qualitatively the geometry of cell segregation depending on the intercellular adhesion parameters. Furthermore, we derive a functional relationship between intercellular adhesion and surface tension and highlight the role of cell mobility in the process of sorting. We show that the interaction between the cells and the boundary of a confining vessel has a major impact on the sorting geometry.

Microenvironmental independence associated with tumor progression

Tumor-microenvironment interactions are increasingly recognized to influence tumor progression. To understand the competitive dynamics of tumor cells in diverse microenvironments, we experimentally parameterized a hybrid discrete-continuum mathematical model with phenotypic trait data from a set of related mammary cell lines with normal, transformed, or tumorigenic properties. Surprisingly, in a resource-rich microenvironment, with few limitations on proliferation or migration, transformed (but not tumorigenic) cells were most successful and outcompeted other cell types in heterogeneous tumor simulations. Conversely, constrained microenvironments with limitations on space and/or growth factors gave a selective advantage to phenotypes derived from tumorigenic cell lines. Analysis of the relative performance of each phenotype in constrained versus unconstrained microenvironments revealed that, although all cell types grew more slowly in resource-constrained microenvironments, the most aggressive cells were least affected by microenvironmental constraints. A game theory model testing the relationship between microenvironment resource availability and competitive cellular dynamics supports the concept that microenvironmental independence is an advantageous cellular trait in resource-limited microenvironments.

Anti-social cells: predicting the influence of E-cadherin loss on the growth of epithelial cell populations

The characteristics of biological tissues are determined by the interactions of large numbers of autonomous cells. These interactions can be mediated remotely by diffusive biochemical factors, or by direct cell-cell contact. E-cadherin is a protein expressed on the surface of normal epithelial cells that plays a key role in mediating intercellular adhesion via calcium-dependent homotypic interactions. E-cadherin is a metastasis-suppressor protein and its loss of function is associated with malignant progression. The purpose of this study was to apply an agent-based simulation paradigm in order to examine the emergent growth properties of mixed populations consisting of normal and E-cadherin defective cells in monolayer cell culture. Specifically, we have investigated the dynamics of normal cell:cell interactions in terms of intercellular adhesion and migration, and have used a simplified rule to represent the concepts of juxtacrine epidermal growth factor receptor (EGFR) activation and subsequent effect on cell proliferation. This cellular level control determines the overall population growth in a simulated experiment. Our approach provides a tool for modelling the development of defined biological abnormalities in epithelial and other biological tissues, raising novel predictions for future experimental testing. The results predict that even a relatively small number of abnormal ('anti-social') cells can modify the rate of the total population expansion, but the magnitude of this effect also depends on the extrinsic (culture) environment. In addition to directly influencing population dynamics, 'anti-social' cells can also disrupt the behaviour of the normal cells around them.

An integrative computational model for intestinal tissue renewal

OBJECTIVES

The luminal surface of the gut is lined with a monolayer of epithelial cells that acts as a nutrient absorptive engine and protective barrier. To maintain its integrity and functionality, the epithelium is renewed every few days. Theoretical models are powerful tools that can be used to test hypotheses concerning the regulation of this renewal process, to investigate how its dysfunction can lead to loss of homeostasis and neoplasia, and to identify potential therapeutic interventions. Here we propose a new multiscale model for crypt dynamics that links phenomena occurring at the subcellular, cellular and tissue levels of organisation.

METHODS

At the subcellular level, deterministic models characterise molecular networks, such as cell-cycle control and Wnt signalling. The output of these models determines the behaviour of each epithelial cell in response to intra-, inter- and extracellular cues. The modular nature of the model enables us to easily modify individual assumptions and analyse their effects on the system as a whole.

RESULTS

We perform virtual microdissection and labelling-index experiments, evaluate the impact of various model extensions, obtain new insight into clonal expansion in the crypt, and compare our predictions with recent mitochondrial DNA mutation data.

CONCLUSIONS

We demonstrate that relaxing the assumption that stem-cell positions are fixed enables clonal expansion and niche succession to occur. We also predict that the presence of extracellular factors near the base of the crypt alone suffices to explain the observed spatial variation in nuclear beta-catenin levels along the crypt axis.