Modeling evolutionary dynamics of epigenetic mutations in hierarchically organized tumors

Minimal cellular automaton model with heterogeneous cell sizes predicts epithelial colony growth

Regulation of cell proliferation is a crucial aspect of tissue development and homeostasis and plays a major role in morphogenesis, wound healing, and tumor invasion. A phenomenon of such regulation is contact inhibition, which describes the dramatic slowing of proliferation, cell migration and individual cell growth when multiple cells are in contact with each other. While many physiological, molecular and genetic factors are known, the mechanism of contact inhibition is still not fully understood. In particular, the relevance of cellular signaling due to interfacial contact for contact inhibition is still debated. Cellular automata (CA) have been employed in the past as numerically efficient mathematical models to study the dynamics of cell ensembles, but they are not suitable to explore the origins of contact inhibition as such agent-based models assume fixed cell sizes. We develop a minimal, data-driven model to simulate the dynamics of planar cell cultures by extending a probabilistic CA to incorporate size changes of individual cells during growth and cell division. We successfully apply this model to previous in-vitro experiments on contact inhibition in epithelial tissue: After a systematic calibration of the model parameters to measurements of single-cell dynamics, our CA model quantitatively reproduces independent measurements of emergent, culture-wide features, like colony size, cell density and collective cell migration. In particular, the dynamics of the CA model also exhibit the transition from a low-density confluent regime to a stationary postconfluent regime with a rapid decrease in cell size and motion. This implies that the volume exclusion principle, a mechanical constraint which is the only inter-cellular interaction incorporated in the model, paired with a size-dependent proliferation rate is sufficient to generate the observed contact inhibition. We discuss how our approach enables the introduction of effective bio-mechanical interactions in a CA framework for future studies.

Evolvability and emergence of tumor heterogeneity as a space-time function

The loss of control of cell proliferation, apoptosis regulation and contact inhibition leads to tumor development. While benign tumors are restricted to their primary space, i.e. where these tumors first originate, the metastatic tumors not only disseminate- facilitated by hypoxia-driven neovascularization- to distant secondary sites but also show substantial changes in metabolism, tissue architectures, gene expression profiles and immune phenotypes. All these alterations result in radio-, chemo- and immune-resistance rendering these metastatic tumor cells refractory to therapy. Since the beginning of the transformation, these factors- which influence each other- are incorporated to the developing and metastasizing tumor. As a result, the complexities in the heterogeneity of tumor progressively increase. This space-time function in the heterogeneity of tumors is generated by various conditions and factors at the genetic as well as microenvironmental levels, for example, endogenous retroviruses, methylation and epigenetic dysregulation that may be etiology-specific, cancer associated inflammation, remodeling of the extracellular matrix and mesenchymal cell shifted functions. On the one hand, these factors may cause de-differentiation of the tumor cells leading to cancer stem cells that contribute to radio-, chemo- and immune-resistance and recurrence of tumors. On the other hand, they may also enhance the heterogeneity under specific microenvironment-driven proliferation. In this editorial, we intend to underline the importance of heterogeneity in cancer progress, its evaluation and its use in correlation with the tumor evolution in a specific patient as a field of research for achieving precise patient-tailored treatments and amelioration of diagnostic (monitoring) tools and prognostic capacity.

Targeting cancer stem cells and signalling pathways through phytochemicals: A promising approach against colorectal cancer

BACKGROUND

Cancer stem cells (CSCs) are strongly associated with high tumourigenicity, chemotherapy or radiotherapy resistance, and metastasis and recurrence, particularly in colorectal cancer (CRC). Therefore, targeting CSCs may be a promising approach. Recently, discovery and research on phytochemicals that effectively target colorectal CSCs have been gaining popularity because of their broad safety profile and multi-target and multi-pathway modes of action.

PURPOSE

This review aimed to elucidate and summarise the effects and mechanisms of phytochemicals with potential anti-CSC agents that could contribute to the better management of CRC.

METHODS

We reviewed PubMed, EMBASE, Web of Science, Ovid, ScienceDirect and China National Knowledge Infrastructure databases from the original publication date to March 2022 to review the mechanisms by which phytochemicals inhibit CRC progression by targeting CSCs and their key signalling pathways. Phytochemicals were classified and summarised based on the mechanisms of action.

RESULTS

We observed that phytochemicals could affect the biological properties of colorectal CSCs. Phytochemicals significantly inhibit self-renewal, migration, invasion, colony formation, and chemoresistance and induce apoptosis and differentiation of CSCs by regulating the Wnt/β-catenin pathway (e.g., diallyl trisulfide and genistein), the phosphatidylinositol-3-kinase/protein kinase B/mammalian target of rapamycin pathway (e.g., caffeic acid and piperlongumine), the neurogenic locus notch homolog protein pathway (e.g., honokiol, quercetin, and α-mangostin), the Janus kinase-signal transducer and activator of transcription pathway (e.g., curcumin, morin, and ursolic acid), and other key signalling pathways. It is worth noting that several phytochemicals, such as resveratrol, silibinin, evodiamine, and thymoquinone, highlight multi-target and multi-pathway effects in restraining the malignant biological behaviour of CSCs.

CONCLUSIONS

This review demonstrates the potential of targeted therapies for colorectal CSCs using phytochemicals. Phytochemicals could serve as novel therapeutic agents for CRC and aid in drug development.

Reconstructing cancer phylogenies using Pairtree, a clone tree reconstruction algorithm

Pairtree is a clone tree reconstruction algorithm that uses somatic point mutations to build clone trees describing the evolutionary history of individual cancers. Using the Pairtree software package, we describe steps to preprocess somatic mutation data, cluster mutations into subclones, search for clone trees, and visualize clone trees. Pairtree builds clone trees using up to 100 samples from a single cancer with at least 30 subclonal populations. For complete details on the use and execution of this protocol, please refer to Wintersinger et al. (2022).

Inferring Tumor Proliferative Organization from Phylogenetic Tree Measures in a Computational Model

We use a computational modeling approach to explore whether it is possible to infer a solid tumor's cellular proliferative hierarchy under the assumptions of the cancer stem cell hypothesis and neutral evolution. We work towards inferring the symmetric division probability for cancer stem cells, since this is believed to be a key driver of progression and therapeutic response. Motivated by the advent of multiregion sampling and resulting opportunities to infer tumor evolutionary history, we focus on a suite of statistical measures of the phylogenetic trees resulting from the tumor's evolution in different regions of parameter space and through time. We find strikingly different patterns in these measures for changing symmetric division probability which hinge on the inclusion of spatial constraints. These results give us a starting point to begin stratifying tumors by this biological parameter and also generate a number of actionable clinical and biological hypotheses regarding changes during therapy, and through tumor evolutionary time. [Cancer; evolution; phylogenetics.].

Effects of mutations and immunogenicity on outcomes of anti-cancer therapies for secondary lesions

Cancer development is driven by mutations and selective forces, including the action of the immune system and interspecific competition. When administered to patients, anti-cancer therapies affect the development and dynamics of tumours, possibly with various degrees of resistance due to immunoediting and microenvironment. Tumours are able to express a variety of competing phenotypes with different attributes and thus respond differently to various anti-cancer therapies. In this paper, a mathematical framework incorporating a system of delay differential equations for the immune system activation cycle and an agent-based approach for tumour-immune interaction is presented. The focus is on those metastatic, secondary solid lesions that are still undetected and non-vascularised. By using available experimental data, we analyse the effects of combination therapies on these lesions and investigate the role of mutations on the rates of success of common treatments. Findings show that mutations, growth properties and immunoediting influence therapies' outcomes in nonlinear and complex ways, affecting cancer lesion morphologies, phenotypical compositions and overall proliferation patterns. Cascade effects on final outcomes for secondary lesions are also investigated, showing that actions on primary lesions could sometimes result in unexpected clearances of secondary tumours. This outcome is strongly dependent on the clonal composition of the primary and secondary masses and is shown to allow, in some cases, the control of the disease for years.

Spatially constrained tumour growth affects the patterns of clonal selection and neutral drift in cancer genomic data

Quantification of the effect of spatial tumour sampling on the patterns of mutations detected in next-generation sequencing data is largely lacking. Here we use a spatial stochastic cellular automaton model of tumour growth that accounts for somatic mutations, selection, drift and spatial constraints, to simulate multi-region sequencing data derived from spatial sampling of a neoplasm. We show that the spatial structure of a solid cancer has a major impact on the detection of clonal selection and genetic drift from both bulk and single-cell sequencing data. Our results indicate that spatial constrains can introduce significant sampling biases when performing multi-region bulk sampling and that such bias becomes a major confounding factor for the measurement of the evolutionary dynamics of human tumours. We also propose a statistical inference framework that incorporates spatial effects within a growing tumour and so represents a further step forwards in the inference of evolutionary dynamics from genomic data. Our analysis shows that measuring cancer evolution using next-generation sequencing while accounting for the numerous confounding factors remains challenging. However, mechanistic model-based approaches have the potential to capture the sources of noise and better interpret the data.

Characterization of Ablation Thresholds for 3D-Cultured Patient-Derived Glioma Stem Cells in Response to High-Frequency Irreversible Electroporation

High-frequency irreversible electroporation (H-FIRE) is a technique that uses pulsed electric fields that have been shown to ablate malignant cells. In order to evaluate the clinical potential of H-FIRE to treat glioblastoma (GBM), a primary brain tumor, we have studied the effects of high-frequency waveforms on therapy-resistant glioma stem-like cell (GSC) populations. We demonstrate that patient-derived GSCs are more susceptible to H-FIRE damage than primary normal astrocytes. This selectivity presents an opportunity for a degree of malignant cell targeting as bulk tumor cells and tumor stem cells are seen to exhibit similar lethal electric field thresholds, significantly lower than that of healthy astrocytes. However, neural stem cell (NSC) populations also exhibit a similar sensitivity to these pulses. This observation may suggest that different considerations be taken when applying these therapies in younger versus older patients, where the importance of preserving NSC populations may impose different restrictions on use. We also demonstrate variability in threshold among the three patient-derived GSC lines studied, suggesting the need for personalized cell-specific characterization in the development of potential clinical procedures. Future work may provide further useful insights regarding this patient-dependent variability observed that could inform targeted and personalized treatment.

Multicompartment cell-based modeling of confined migration: regulation by cell intrinsic and extrinsic factors

Though cell and nuclear deformability are expected to influence efficiency of confined migration, their individual and collective influence on migration efficiency remains incompletely understood. In addition to cell intrinsic properties, the relevance of cell extrinsic factors on confined migration, if any, has not been adequately explored. Here we address these questions using a statistical mechanics-based stochastic modeling approach where cell/nuclear dimensions and their deformability are explicitly taken into consideration. In addition to demonstrating the importance of cell softness in sustaining confined migration, our results suggest that dynamic tuning of cell and nuclear properties at different stages of migration is essential for maximizing migration efficiency. Our simulations also implicate confinement shape and confinement history as two important cell extrinsic regulators of cell invasiveness. Together, our findings illustrate the strength of a multicompartment model in dissecting the contributions of multiple factors that collectively influence confined cell migration.

Heritable tumor cell division rate heterogeneity induces clonal dominance

Tumors consist of a hierarchical population of cells that differ in their phenotype and genotype. This hierarchical organization of cells means that a few clones (i.e., cells and several generations of offspring) are abundant while most are rare, which is called clonal dominance. Such dominance also occurred in published in vitro iterated growth and passage experiments with tumor cells in which genetic barcodes were used for lineage tracing. A potential source for such heterogeneity is that dominant clones derive from cancer stem cells with an unlimited self-renewal capacity. Furthermore, ongoing evolution and selection within the growing population may also induce clonal dominance. To understand how clonal dominance developed in the iterated growth and passage experiments, we built a computational model that accurately simulates these experiments. The model simulations reproduced the clonal dominance that developed in in vitro iterated growth and passage experiments when the division rates vary between cells, due to a combination of initial variation and of ongoing mutational processes. In contrast, the experimental results can neither be reproduced with a model that considers random growth and passage, nor with a model based on cancer stem cells. Altogether, our model suggests that in vitro clonal dominance develops due to selection of fast-dividing clones.

Tumors consist of numerous cell populations, i.e., clones, that differ with respect to genotype, and potentially with respect to phenotype, and these populations strongly differ in their size. A limited number of clones tend to dominate tumors, but it remains unclear how this clonal dominance arises. Potential driving mechanisms are the presence of cancer stem cells, which either divide indefinitely of differentiate into cells with a limited division potential, and ongoing evolutionary processes within the tumor. Here we use a computational model to understand how clonal dominance developed during in vitro growth and passage experiments with cancer cells. Incorporating cancer stem cells in this model did not result in a match between simulations and in vitro data. In contrast, by considering all cells to divide indefinitely and division rates to evolve due to the combination of division rate variability and selection by passage, our model closely matches the in vitro data.

Agent-Based Modeling of Cancer Stem Cell Driven Solid Tumor Growth

Computational modeling of tumor growth has become an invaluable tool to simulate complex cell-cell interactions and emerging population-level dynamics. Agent-based models are commonly used to describe the behavior and interaction of individual cells in different environments. Behavioral rules can be informed and calibrated by in vitro assays, and emerging population-level dynamics may be validated with both in vitro and in vivo experiments. Here, we describe the design and implementation of a lattice-based agent-based model of cancer stem cell driven tumor growth.

Catch my drift? Making sense of genomic intra-tumour heterogeneity

The cancer genome is shaped by three components of the evolutionary process: mutation, selection and drift. While many studies have focused on the first two components, the role of drift in cancer evolution has received little attention. Drift occurs when all individuals in the population have the same likelihood of producing surviving offspring, and so by definition a drifting population is one that is evolving neutrally. Here we focus on how neutral evolution is manifested in the cancer genome. We discuss how neutral passenger mutations provide a magnifying glass that reveals the evolutionary dynamics underpinning cancer development, and outline how statistical inference can be used to quantify these dynamics from sequencing data. We argue that only after we understand the impact of neutral drift on the genome can we begin to make full sense of clonal selection. This article is part of a Special Issue entitled: Evolutionary principles - heterogeneity in cancer? Edited by Dr. Robert A. Gatenby.

Addiction to the IGF2-ID1-IGF2 circuit for maintenance of the breast cancer stem-like cells

The transcription factor nuclear factor-κB (NF-κB) has important roles for tumorigenesis, but how it regulates cancer stem cells (CSCs) remains largely unclear. We identified insulin-like growth factor 2 (IGF2) is a key target of NF-κB activated by HER2/HER3 signaling to form tumor spheres in breast cancer cells. The IGF2 receptor, IGF1 R, was expressed at high levels in CSC-enriched populations in primary breast cancer cells. Moreover, IGF2-PI3K (IGF2-phosphatidyl inositol 3 kinase) signaling induced expression of a stemness transcription factor, inhibitor of DNA-binding 1 (ID1), and IGF2 itself. ID1 knockdown greatly reduced IGF2 expression, and tumor sphere formation. Finally, treatment with anti-IGF1/2 antibodies blocked tumorigenesis derived from the IGF1R^high CSC-enriched population in a patient-derived xenograft model. Thus, NF-κB may trigger IGF2-ID1-IGF2-positive feedback circuits that allow cancer stem-like cells to appear. Then, they may become addicted to the circuits. As the circuits are the Achilles' heels of CSCs, it will be critical to break them for eradication of CSCs.

In silico modeling for tumor growth visualization

BACKGROUND

Cancer is a complex disease. Fundamental cellular based studies as well as modeling provides insight into cancer biology and strategies to treatment of the disease. In silico models complement in vivo models. Research on tumor growth involves a plethora of models each emphasizing isolated aspects of benign and malignant neoplasms. Biologists and clinical scientists are often overwhelmed by the mathematical background knowledge necessary to grasp and to apply a model to their own research.

RESULTS

We aim to provide a comprehensive and expandable simulation tool to visualizing tumor growth. This novel Web-based application offers the advantage of a user-friendly graphical interface with several manipulable input variables to correlate different aspects of tumor growth. By refining model parameters we highlight the significance of heterogeneous intercellular interactions on tumor progression. Within this paper we present the implementation of the Cellular Potts Model graphically presented through Cytoscape.js within a Web application. The tool is available under the MIT license at https://github.com/davcem/cpm-cytoscape and http://styx.cgv.tugraz.at:8080/cpm-cytoscape/ .

CONCLUSION

In-silico methods overcome the lack of wet experimental possibilities and as dry method succeed in terms of reduction, refinement and replacement of animal experimentation, also known as the 3R principles. Our visualization approach to simulation allows for more flexible usage and easy extension to facilitate understanding and gain novel insight. We believe that biomedical research in general and research on tumor growth in particular will benefit from the systems biology perspective.

Feedback Regulation in a Cancer Stem Cell Model can Cause an Allee Effect

The exact mechanisms of spontaneous tumor remission or complete response to treatment are phenomena in oncology that are not completely understood. We use a concept from ecology, the Allee effect, to help explain tumor extinction in a model of tumor growth that incorporates feedback regulation of stem cell dynamics, which occurs in many tumor types where certain signaling molecules, such as Wnts, are upregulated. Due to feedback and the Allee effect, a tumor may become extinct spontaneously or after therapy even when the entire tumor has not been eradicated by the end of therapy. We quantify the Allee effect using an 'Allee index' that approximates the area of the basin of attraction for tumor extinction. We show that effectiveness of combination therapy in cancer treatment may occur due to the increased probability that the system will be in the Allee region after combination treatment versus monotherapy. We identify therapies that can attenuate stem cell self-renewal, alter the Allee region and increase its size. We also show that decreased response of tumor cells to growth inhibitors can reduce the size of the Allee region and increase stem cell densities, which may help to explain why this phenomenon is a hallmark of cancer.

Bidirectional interconversion of stem and non-stem cancer cell populations: A reassessment of theoretical models for tumor heterogeneity

Resolving the origin of intratumor heterogeneity has proven to be one of the central challenges in cancer research during recent years. Two theoretical models explaining the emergence of intratumor heterogeneity have come to dominate cancer biology literature: the clonal evolution model and the hierarchical/cancer stem cell model. Recently, a plastic model that combines elements of both the clonal and the hierarchical model has gained traction. Basically, this model proposes that cancer stem cells engage in bidirectional interconversion with non-stem cells, thereby providing the missing link between the 2 conventional models. Confirming bidirectional interconversion as a hallmark of cancer is a crucial step in understanding tumor heterogeneity and has important therapeutic implications. In this review, current methodologies and theoretical and empirical evidence regarding bidirectional interconversion will be discussed.

Simulation-based Bayesian Analysis of Complex Data

Our ability to collect large datasets is growing rapidly. Such richness of data offers great promise in terms of addressing detailed scientific questions in great depth. However, this benefit is not without scientific difficulty: many traditional analysis methods become computationally intractable for very large datasets. However, one can frequently still simulate data from scientific models for which direct calculation is no longer possible. In this paper we propose a Bayesian perspective for such analyses, and argue for the advantage of a simulation-based approximate Bayesian method that remains tractable when tractability of other methods is lost. This method, which is known as "approximate Bayesian computation" [ABC], has now been used in a variety of contexts, such as the analysis of tumor data (a tumor being a complex population of cells), and the analysis of human genetic variation data (which arise from a population of individual people). We review a number of ABC methods, with specific attention to the use of ABC in agent-based models, and give pointers to software that allows straightforward implementation of the ABC approach. In this way we demonstrate the utility of simulation-based analyses of large datasets within a rigorous statistical framework.

The role of the bi-compartmental stem cell niche in delaying cancer

In recent years, by using modern imaging techniques, scientists have found evidence of collaboration between different types of stem cells (SCs), and proposed a bi-compartmental organization of the SC niche. Here we create a class of stochastic models to simulate the dynamics of such a heterogeneous SC niche. We consider two SC groups: the border compartment, S1, is in direct contact with transit-amplifying (TA) cells, and the central compartment, S2, is hierarchically upstream from S1. The S1 SCs differentiate or divide asymmetrically when the tissue needs TA cells. Both groups proliferate when the tissue requires SCs (thus maintaining homeostasis). There is an influx of S2 cells into the border compartment, either by migration, or by proliferation. We examine this model in the context of double-hit mutant generation, which is a rate-limiting step in the development of many cancers. We discover that this type of a cooperative pattern in the stem niche with two compartments leads to a significantly smaller rate of double-hit mutant production compared with a homogeneous, one-compartmental SC niche. Furthermore, the minimum probability of double-hit mutant generation corresponds to purely symmetric division of SCs, consistent with the literature. Finally, the optimal architecture (which minimizes the rate of double-hit mutant production) requires a large proliferation rate of S1 cells along with a small, but non-zero, proliferation rate of S2 cells. This result is remarkably similar to the niche structure described recently by several authors, where one of the two SC compartments was found more actively engaged in tissue homeostasis and turnover, while the other was characterized by higher levels of quiescence (but contributed strongly to injury recovery). Both numerical and analytical results are presented.

Colorectal cancer stem cells: from the crypt to the clinic

Since their first discovery, investigations of colorectal cancer stem cells (CSCs) have revealed some unexpected properties, including a high degree of heterogeneity and plasticity. By exploiting a combination of genetic, epigenetic, and microenvironmental factors, colorectal CSCs metastasize, resist chemotherapy, and continually adapt to a changing microenvironment, representing a formidable challenge to cancer eradication. Here, we review the current understanding of colorectal CSCs, including their origin, relationship to stem cells of the intestine, phenotypic characterization, and underlying regulatory mechanisms. We also discuss limitations to current preclinical models of colorectal cancer and how understanding CSC plasticity can improve the development of clinical strategies.

An agent-based model of cancer stem cell initiated avascular tumour growth and metastasis: the effect of seeding frequency and location

It is very important to understand the onset and growth pattern of breast primary tumours as well as their metastatic dissemination. In most cases, it is the metastatic disease that ultimately kills the patient. There is increasing evidence that cancer stem cells are closely linked to the progression of the metastatic tumour. Here, we investigate stem cell seeding to an avascular tumour site using an agent-based stochastic model of breast cancer metastatic seeding. The model includes several important cellular features such as stem cell symmetric and asymmetric division, migration, cellular quiescence, senescence, apoptosis and cell division cycles. It also includes external features such as stem cell seeding frequency and location. Using this model, we find that cell seeding rate and location are important features for tumour growth. We also define conditions in which the tumour growth exhibits decremented and exponential growth patterns. Overall, we find that seeding, senescence and division limit affect not only the number of stem cells, but also their spatial and temporal distribution.

Heterogeneity of chemokine cell-surface receptor expression in triple-negative breast cancer

INTRODUCTION

Tumor heterogeneity is a well-established concept in cancer research. In this paper, we examine an additional type of tumor cell heterogeneity - tumor cell-surface receptor heterogeneity.

METHODS

We use flow cytometry to measure the frequency and numbers of cell-surface receptors on triple negative breast cancer cell lines.

RESULTS

We find two distinct populations of human triple-negative breast cancer cells MDA-MB-231 when they are grown in culture, one with low surface levels of various chemokine receptors and a second with much higher levels. The population with high surface levels of these receptors is increased in the more metastatic MDA-MB-231-luc-d3h2ln cell line.

CONCLUSION

We hypothesize that this high cell-surface receptor population is involved in metastasis. We find that the receptor high populations can be modulated by tumor conditioned media and IL6 treatment indicating that the tumor microenvironment is important for the maintenance and sizes of these populations.

Cancer stem cells: small subpopulation or evolving fraction?

This review discusses quantitative modeling studies of stem and non-stem cancer cell interactions and the fraction of cancer stem cells.

Cancer: evolution within a lifetime

Subclonal cancer populations change spatially and temporally during the disease course. Studies are revealing branched evolutionary cancer growth with low-frequency driver events present in subpopulations of cells, providing escape mechanisms for targeted therapeutic approaches. Despite such complexity, evidence is emerging for parallel evolution of subclones, mediated through distinct somatic events converging on the same gene, signal transduction pathway, or protein complex in different subclones within the same tumor. Tumors may follow gradualist paths (microevolution) as well as major shifts in evolutionary trajectories (macroevolution). Although macroevolution has been subject to considerable controversy in post-Darwinian evolutionary theory, we review evidence that such nongradual, saltatory leaps, driven through chromosomal rearrangements or genome doubling, may be particularly relevant to tumor evolution. Adapting cancer care to the challenges imposed by tumor micro- and macroevolution and developing deeper insight into parallel evolutionary events may prove central to improving outcome and reducing drug development costs.

Exploiting ecological principles to better understand cancer progression and treatment

A small but growing number of people are finding interesting parallels between ecosystems as studied by ecologists (think of a savannah or the Amazon rainforest or a coral reef) and tumours. The idea of viewing cancer from an ecological perspective has many implications but, basically, it means that we should not see cancer just as a group of mutated cells. A more useful definition of cancer is to consider it a disruption in the complex balance of many interacting cellular and microenvironmental elements in a specific organ. This perspective means that organs undergoing carcinogenesis should be seen as sophisticated ecosystems in homoeostasis that cancer cells can disrupt. It also makes cancer seem even more complex but may ultimately provide insights that make it more treatable. Here, we discuss how ecological principles can be used to better understand cancer progression and treatment, using several mathematical and computational models to illustrate our argument.

From computational modelling of the intrinsic apoptosis pathway to a systems-based analysis of chemotherapy resistance: achievements, perspectives and challenges in systems medicine

Our understanding of the mitochondrial or intrinsic apoptosis pathway and its role in chemotherapy resistance has increased significantly in recent years by a combination of experimental studies and mathematical modelling. This combined approach enhanced the quantitative and kinetic understanding of apoptosis signal transduction, but also provided new insights that systems-emanating functions (i.e., functions that cannot be attributed to individual network components but that are instead established by multi-component interplay) are crucial determinants of cell fate decisions. Among these features are molecular thresholds, cooperative protein functions, feedback loops and functional redundancies that provide systems robustness, and signalling topologies that allow ultrasensitivity or switch-like responses. The successful development of kinetic systems models that recapitulate biological signal transduction observed in living cells have now led to the first translational studies, which have exploited and validated such models in a clinical context. Bottom-up strategies that use pathway models in combination with higher-level modelling at the tissue, organ and whole body-level therefore carry great potential to eventually deliver a new generation of systems-based diagnostic tools that may contribute to the development of personalised and predictive medicine approaches. Here we review major achievements in the systems biology of intrinsic apoptosis signalling, discuss challenges for further model development, perspectives for higher-level integration of apoptosis models and finally discuss requirements for the development of systems medical solutions in the coming years.

A multi-phenotypic cancer model with cell plasticity

The conventional cancer stem cell (CSC) theory indicates a hierarchy of CSCs and non-stem cancer cells (NSCCs), that is, CSCs can differentiate into NSCCs but not vice versa. However, an alternative paradigm of CSC theory with reversible cell plasticity among cancer cells has received much attention very recently. Here we present a generalized multi-phenotypic cancer model by integrating cell plasticity with the conventional hierarchical structure of cancer cells. We prove that under very weak assumption, the nonlinear dynamics of multi-phenotypic proportions in our model has only one stable steady state and no stable limit cycle. This result theoretically explains the phenotypic equilibrium phenomena reported in various cancer cell lines. Furthermore, according to the transient analysis of our model, it is found that cancer cell plasticity plays an essential role in maintaining the phenotypic diversity in cancer especially during the transient dynamics. Two biological examples with experimental data show that the phenotypic conversions from NCSSs to CSCs greatly contribute to the transient growth of CSCs proportion shortly after the drastic reduction of it. In particular, an interesting overshooting phenomenon of CSCs proportion arises in three-phenotypic example. Our work may pave the way for modeling and analyzing the multi-phenotypic cell population dynamics with cell plasticity.

Evolution of intratumoral phenotypic heterogeneity: the role of trait inheritance

A tumour is a heterogeneous population of cells that competes for limited resources. In the clinic, we typically probe the tumour by biopsy, and then characterize it by the dominant genetic clone. But genotypes are only the first link in the chain of hierarchical events that leads to a specific cell phenotype. The relationship between genotype and phenotype is not simple, and the so-called genotype to phenotype map is poorly understood. Many genotypes can produce the same phenotype, so genetic heterogeneity may not translate directly to phenotypic heterogeneity. We therefore choose to focus on the functional endpoint, the phenotype as defined by a collection of cellular traits (e.g. proliferative and migratory ability). Here, we will examine how phenotypic heterogeneity evolves in space and time and how the way in which phenotypes are inherited will drive this evolution. A tumour can be thought of as an ecosystem, which critically means that we cannot just consider it as a collection of mutated cells but more as a complex system of many interacting cellular and microenvironmental elements. At its simplest, a growing tumour with increased proliferation capacity must compete for space as a limited resource. Hypercellularity leads to a contact-inhibited core with a competitive proliferating rim. Evolution and selection occurs, and an individual cell's capacity to survive and propagate is determined by its combination of traits and interaction with the environment. With heterogeneity in phenotypes, the clone that will dominate is not always obvious as there are both local interactions and global pressures. Several combinations of phenotypes can coexist, changing the fitness of the whole. To understand some aspects of heterogeneity in a growing tumour, we build an off-lattice agent-based model consisting of individual cells with assigned trait values for proliferation and migration rates. We represent heterogeneity in these traits with frequency distributions and combinations of traits with density maps. How the distributions change over time is dependent on how traits are passed on to progeny cells, which is our main enquiry. We bypass the translation of genetics to behaviour by focusing on the functional end result of inheritance of the phenotype combined with the environmental influence of limited space.

Population dynamics of cancer cells with cell state conversions

Cancer stem cell (CSC) theory suggests a cell-lineage structure in tumor cells in which CSCs are capable of giving rise to the other non-stem cancer cells (NSCCs) but not vice versa. However, an alternative scenario of bidirectional interconversions between CSCs and NSCCs was proposed very recently. Here we present a general population model of cancer cells by integrating conventional cell divisions with direct conversions between different cell states, namely, not only can CSCs differentiate into NSCCs by asymmetric cell division, NSCCs can also dedifferentiate into CSCs by cell state conversion. Our theoretical model is validated when applying the model to recent experimental data. It is also found that the transient increase in CSCs proportion initiated from the purified NSCCs subpopulation cannot be well predicted by the conventional CSC model where the conversion from NSCCs to CSCs is forbidden, implying that the cell state conversion is required especially for the transient dynamics. The theoretical analysis also gives the condition such that our general model can be equivalently reduced into a simple Markov chain with only cell state transitions keeping the same cell proportion dynamics.

Cellular potts modeling of tumor growth, tumor invasion, and tumor evolution

Despite a growing wealth of available molecular data, the growth of tumors, invasion of tumors into healthy tissue, and response of tumors to therapies are still poorly understood. Although genetic mutations are in general the first step in the development of a cancer, for the mutated cell to persist in a tissue, it must compete against the other, healthy or diseased cells, for example by becoming more motile, adhesive, or multiplying faster. Thus, the cellular phenotype determines the success of a cancer cell in competition with its neighbors, irrespective of the genetic mutations or physiological alterations that gave rise to the altered phenotype. What phenotypes can make a cell "successful" in an environment of healthy and cancerous cells, and how? A widely used tool for getting more insight into that question is cell-based modeling. Cell-based models constitute a class of computational, agent-based models that mimic biophysical and molecular interactions between cells. One of the most widely used cell-based modeling formalisms is the cellular Potts model (CPM), a lattice-based, multi particle cell-based modeling approach. The CPM has become a popular and accessible method for modeling mechanisms of multicellular processes including cell sorting, gastrulation, or angiogenesis. The CPM accounts for biophysical cellular properties, including cell proliferation, cell motility, and cell adhesion, which play a key role in cancer. Multiscale models are constructed by extending the agents with intracellular processes including metabolism, growth, and signaling. Here we review the use of the CPM for modeling tumor growth, tumor invasion, and tumor progression. We argue that the accessibility and flexibility of the CPM, and its accurate, yet coarse-grained and computationally efficient representation of cell and tissue biophysics, make the CPM the method of choice for modeling cellular processes in tumor development.

Intratumor heterogeneity in human glioblastoma reflects cancer evolutionary dynamics

Glioblastoma (GB) is the most common and aggressive primary brain malignancy, with poor prognosis and a lack of effective therapeutic options. Accumulating evidence suggests that intratumor heterogeneity likely is the key to understanding treatment failure. However, the extent of intratumor heterogeneity as a result of tumor evolution is still poorly understood. To address this, we developed a unique surgical multisampling scheme to collect spatially distinct tumor fragments from 11 GB patients. We present an integrated genomic analysis that uncovers extensive intratumor heterogeneity, with most patients displaying different GB subtypes within the same tumor. Moreover, we reconstructed the phylogeny of the fragments for each patient, identifying copy number alterations in EGFR and CDKN2A/B/p14ARF as early events, and aberrations in PDGFRA and PTEN as later events during cancer progression. We also characterized the clonal organization of each tumor fragment at the single-molecule level, detecting multiple coexisting cell lineages. Our results reveal the genome-wide architecture of intratumor variability in GB across multiple spatial scales and patient-specific patterns of cancer evolution, with consequences for treatment design.

Determining duration of HER2-targeted therapy using stem cell extinction models

Introduction

Trastuzumab dramatically improves survival in breast cancer patients whose tumor overexpresses HER2. A subpopulation of cells in human breast tumors has been identified with characteristics of cancer stem cells. These breast cancer stem-like cells (BCSCs) rely on HER2 signaling for self-renewal, suggesting that HER2-targeted therapy targets BCSCs even when the bulk of the tumor does not overexpress HER2. In order to guide clinical trials examining HER2-targeted therapy in the adjuvant setting, we propose a mathematical model to examine BCSC population dynamics and predict optimal duration of therapy.

Methods

Varying the susceptibility of BCSCs to HER2-targeted therapy, we quantify the average time to extinction of BCSCs. We expand our model using stochastic simulation to include the partially differentiated tumor cells (TCs) that represent bulk tumor population and examine effects of plasticity on required duration of therapy.

Results

Lower susceptibility of BCSCs and increased rates of dedifferentiation entail longer extinction times, indicating a need for prolonged administration of HER2-targeted therapy. We predict that even when therapy does not appreciably reduce tumor size in the advanced cancer setting, it will eventually eradicate the tumor in the adjuvant setting as long as there is at least a modest effect on BCSCs.

Conclusions

We anticipate that our results will inform clinical trials of targeted therapies in planning the duration of therapy needed to eradicate BCSCs. Our predictions also address safety, as longer duration of therapy entails a greater potential impact on normal stem cells that may also be susceptible to stem cell-targeted therapies.

Applying ecological and evolutionary theory to cancer: a long and winding road

Since the mid 1970s, cancer has been described as a process of Darwinian evolution, with somatic cellular selection and evolution being the fundamental processes leading to malignancy and its many manifestations (neoangiogenesis, evasion of the immune system, metastasis, and resistance to therapies). Historically, little attention has been placed on applications of evolutionary biology to understanding and controlling neoplastic progression and to prevent therapeutic failures. This is now beginning to change, and there is a growing international interest in the interface between cancer and evolutionary biology. The objective of this introduction is first to describe the basic ideas and concepts linking evolutionary biology to cancer. We then present four major fronts where the evolutionary perspective is most developed, namely laboratory and clinical models, mathematical models, databases, and techniques and assays. Finally, we discuss several of the most promising challenges and future prospects in this interdisciplinary research direction in the war against cancer.

Single-molecule genomic data delineate patient-specific tumor profiles and cancer stem cell organization

Substantial evidence supports the concept that cancers are organized in a cellular hierarchy with cancer stem cells (CSC) at the apex. To date, the primary evidence for CSCs derives from transplantation assays, which have known limitations. In particular, they are unable to report on the fate of cells within the original human tumor. Because of the difficulty in measuring tumor characteristics in patients, cellular organization and other aspects of cancer dynamics have not been quantified directly, although they likely play a fundamental role in tumor progression and therapy response. As such, new approaches to study CSCs in patient-derived tumor specimens are needed. In this study, we exploited ultradeep single-molecule genomic data derived from multiple microdissected colorectal cancer glands per tumor, along with a novel quantitative approach to measure tumor characteristics, define patient-specific tumor profiles, and infer tumor ancestral trees. We show that each cancer is unique in terms of its cellular organization, molecular heterogeneity, time from malignant transformation, and rate of mutation and apoptosis. Importantly, we estimate CSC fractions between 0.5% and 4%, indicative of a hierarchical organization responsible for long-lived CSC lineages, with variable rates of symmetric cell division. We also observed extensive molecular heterogeneity, both between and within individual cancer glands, suggesting a complex hierarchy of mitotic clones. Our framework enables the measurement of clinically relevant patient-specific characteristics in vivo, providing insight into the cellular organization and dynamics of tumor growth, with implications for personalized patient care.

Riparian ecosystems in human cancers

Intratumoral evolution produces extensive genetic heterogeneity in clinical cancers. This is generally attributed to an increased mutation rate that continually produces new genetically defined clonal lineages. Equally important are the interactions between the heritable traits of cancer cells and their microenvironment that produces natural selection favoring some clonal ‘species’ over others. That is, while mutations produce the heritable variation, environmental selection and cellular adaptation govern the strategies (and genotypes) that can proliferate within the tumor ecosystem. Here we ask: What are the dominant evolutionary forces in the cancer ecosystem? We propose that the tumor vascular network is a common and primary cause of intratumoral heterogeneity. Specifically, variations in blood flow result in variability in substrate, such as oxygen, and metabolites, such as acid, that serve as critical, but predictable, environmental selection forces. We examine the evolutionary and ecological consequences of variable blood flow by drawing an analogy to riparian habitats within desert landscapes. We propose that the phenotypic properties of cancer cells will exhibit predictable spatial variation within tumor phenotypes as a result of proximity to blood flow. Just as rivers in the desert create an abrupt shift from the lush, mesic riparian vegetation along the banks to sparser, xeric and dry-adapted plant species in the adjacent drylands, we expect blood vessels within tumors to promote similarly distinct communities of cancer cells that change abruptly with distance from the blood vessel. We propose vascular density and blood flow within a tumor as a primary evolutionary force governing variations in the phenotypic properties of cancer cells thus providing a unifying ecological framework for understanding intratumoral heterogeneity.

Multi-scale modeling in morphogenesis: a critical analysis of the cellular Potts model

Cellular Potts models (CPMs) are used as a modeling framework to elucidate mechanisms of biological development. They allow a spatial resolution below the cellular scale and are applied particularly when problems are studied where multiple spatial and temporal scales are involved. Despite the increasing usage of CPMs in theoretical biology, this model class has received little attention from mathematical theory. To narrow this gap, the CPMs are subjected to a theoretical study here. It is asked to which extent the updating rules establish an appropriate dynamical model of intercellular interactions and what the principal behavior at different time scales characterizes. It is shown that the longtime behavior of a CPM is degenerate in the sense that the cells consecutively die out, independent of the specific interdependence structure that characterizes the model. While CPMs are naturally defined on finite, spatially bounded lattices, possible extensions to spatially unbounded systems are explored to assess to which extent spatio-temporal limit procedures can be applied to describe the emergent behavior at the tissue scale. To elucidate the mechanistic structure of CPMs, the model class is integrated into a general multiscale framework. It is shown that the central role of the surface fluctuations, which subsume several cellular and intercellular factors, entails substantial limitations for a CPM's exploitation both as a mechanistic and as a phenomenological model.

In vitro models

The current resurgence of interest in the cancer stem cell (CSC) hypothesis as possibly providing a unifying theory of cancer biology is fueled by the growing body of work on normal adult tissue stem cells and the promise that CSC may hold the key to one of the central problems of clinical oncology: tumor recurrence. Many studies suggest that the microenvironment plays a role, perhaps a seminal one, in cancer development and progression. In addition, the possibility that the stem cell-like component of tumors is capable of rapid and reversible changes of phenotype raises questions concerning studies with these populations and the application of what we learn to the clinical situation. These types of questions are extremely difficult to study using in vivo models or freshly isolated cells. Established cell lines grown in defined conditions provide important model systems for these studies. There are three types of in vitro models for CSCs: (a) selected subpopulations of existing tumor lines (derived from serum-containing medium; (b) creation of lines from tumor or normal cells by genetic manipulation; or (c) direct in vitro selection of CSC from tumors or sorted tumor cells using defined serum-free conditions. We review the problems associated with creating and maintaining in vitro cultures of CSCs and the progress to date on the establishment of these important models.

The developing cancer stem-cell model: clinical challenges and opportunities

During the past decade, a stem-cell-like subset of cancer cells has been identified in many malignancies. These cells, referred to as cancer stem cells (CSCs), are of particular interest because they are believed to be the clonogenic core of the tumour and therefore represent the cell population that drives growth and progression. Many efforts have been made to design therapies that specifically target the CSC population, since this was predicted to be the crucial population to eliminate. However, recent insights have complicated the initial elegant model, by showing a dominant role for the tumour microenvironment in determining CSC characteristics within a malignancy. This is particularly important since dedifferentiation of non-tumorigenic tumour cells towards CSCs can occur, and therefore the CSC population in a neoplasm is expected to vary over time. Moreover, evidence suggests that not all tumours are driven by rare CSCs, but might instead contain a large population of tumorigenic cells. Even though these results suggest that specific targeting of the CSC population might not be a useful therapeutic strategy, research into the hierarchical cellular organisation of malignancies has provided many important new insights in the biology of tumours. In this Personal View, we highlight how the CSC concept is developing and influences our thinking on future treatment for solid tumours, and recommend ways to design clinical trials to assess drugs that target malignant disease in a rational fashion.

Efficient coarse simulation of a growing avascular tumor

The subject of this work is the development and implementation of algorithms which accelerate the simulation of early stage tumor growth models. Among the different computational approaches used for the simulation of tumor progression, discrete stochastic models (e.g., cellular automata) have been widely used to describe processes occurring at the cell and subcell scales (e.g., cell-cell interactions and signaling processes). To describe macroscopic characteristics (e.g., morphology) of growing tumors, large numbers of interacting cells must be simulated. However, the high computational demands of stochastic models make the simulation of large-scale systems impractical. Alternatively, continuum models, which can describe behavior at the tumor scale, often rely on phenomenological assumptions in place of rigorous upscaling of microscopic models. This limits their predictive power. In this work, we circumvent the derivation of closed macroscopic equations for the growing cancer cell populations; instead, we construct, based on the so-called "equation-free" framework, a computational superstructure, which wraps around the individual-based cell-level simulator and accelerates the computations required for the study of the long-time behavior of systems involving many interacting cells. The microscopic model, e.g., a cellular automaton, which simulates the evolution of cancer cell populations, is executed for relatively short time intervals, at the end of which coarse-scale information is obtained. These coarse variables evolve on slower time scales than each individual cell in the population, enabling the application of forward projection schemes, which extrapolate their values at later times. This technique is referred to as coarse projective integration. Increasing the ratio of projection times to microscopic simulator execution times enhances the computational savings. Crucial accuracy issues arising for growing tumors with radial symmetry are addressed by applying the coarse projective integration scheme in a cotraveling (cogrowing) frame. As a proof of principle, we demonstrate that the application of this scheme yields highly accurate solutions, while preserving the computational savings of coarse projective integration.