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

Lattice-based model of ductal carcinoma in situ suggests rules for breast cancer progression to an invasive state

Ductal carcinoma in situ (DCIS) is a heterogeneous group of non-invasive lesions of the breast that result from abnormal proliferation of mammary epithelial cells. Pathologists characterize DCIS by four tissue morphologies (micropapillary, cribriform, solid, and comedo), but the underlying mechanisms that distinguish the development and progression of these morphologies are not well understood. Here we explored the conditions leading to the emergence of the different morphologies of DCIS using a two-dimensional multi-cell lattice-based model that incorporates cell proliferation, apoptosis, necrosis, adhesion, and contractility. We found that the relative rates of cell proliferation and apoptosis governed which of the four morphologies emerged. High proliferation and low apoptosis favored the emergence of solid and comedo morphologies. In contrast, low proliferation and high apoptosis led to the micropapillary morphology, whereas high proliferation and high apoptosis led to the cribriform morphology. The natural progression between morphologies cannot be investigated in vivo since lesions are usually surgically removed upon detection; however, our model suggests probable transitions between these morphologies during breast cancer progression. Importantly, cribriform and comedo appear to be the ultimate morphologies of DCIS. Motivated by previous experimental studies demonstrating that tumor cells behave differently depending on where they are located within the mammary duct in vivo or in engineered tissues, we examined the effects of tissue geometry on the progression of DCIS. In agreement with our previous experimental work, we found that cells are more likely to invade from the end of ducts and that this preferential invasion is regulated by cell adhesion and contractility. This model provides additional insight into tumor cell behavior and allows the exploration of phenotypic transitions not easily monitored in vivo.

Breast cancer is a complex disease that affects women worldwide. One heterogeneous group of lesions, ductal carcinoma in situ (DCIS), often begins as a nonmalignant disease but can readily progress if left untreated. The progression of this disease is not well understood because DCIS is typically removed upon detection. Therefore, computational models might help predict whether DCIS will remain nonmalignant or progress towards invasive ductal carcinoma. Here we used a multi-cell lattice-based model to explore the relative effects of cell proliferation, death, division axis, adhesion and contractility on the development and progression of DCIS. We also examined the emergence and progression of DCIS in physiologically relevant geometries of the mammary duct. Our model suggests several plausible progressions between morphologies of DCIS, and predicts that some regions of a duct are preferential for tumor cell invasion.

Computational analysis of three-dimensional epithelial morphogenesis using vertex models

The folding of epithelial sheets, accompanied by cell shape changes and rearrangements, gives rise to three-dimensional structures during development. Recently, some aspects of epithelial morphogenesis have been modeled using vertex models, in which each cell is approximated by a polygon; however, these models have been largely confined to two dimensions. Here, we describe an adaptation of these models in which the classical two-dimensional vertex model is embedded in three dimensions. This modification allows for the construction of complex three-dimensional shapes from simple sheets of cells. We describe algorithmic, computational, and biophysical aspects of our model, with the view that it may be useful for formulating and testing hypotheses regarding the mechanical forces underlying a wide range of morphogenetic processes.

Evolutionary game theory for physical and biological scientists. I. Training and validating population dynamics equations

Failure to understand evolutionary dynamics has been hypothesized as limiting our ability to control biological systems. An increasing awareness of similarities between macroscopic ecosystems and cellular tissues has inspired optimism that game theory will provide insights into the progression and control of cancer. To realize this potential, the ability to compare game theoretic models and experimental measurements of population dynamics should be broadly disseminated. In this tutorial, we present an analysis method that can be used to train parameters in game theoretic dynamics equations, used to validate the resulting equations, and used to make predictions to challenge these equations and to design treatment strategies. The data analysis techniques in this tutorial are adapted from the analysis of reaction kinetics using the method of initial rates taught in undergraduate general chemistry courses. Reliance on computer programming is avoided to encourage the adoption of these methods as routine bench activities.

Evolutionary game theory for physical and biological scientists. II. Population dynamics equations can be associated with interpretations

The use of mathematical equations to analyse population dynamics measurements is being increasingly applied to elucidate complex dynamic processes in biological systems, including cancer. Purely 'empirical' equations may provide sufficient accuracy to support predictions and therapy design. Nevertheless, interpretation of fitting equations in terms of physical and biological propositions can provide additional insights that can be used both to refine models that prove inconsistent with data and to understand the scope of applicability of models that validate. The purpose of this tutorial is to assist readers in mathematically associating interpretations with equations and to provide guidance in choosing interpretations and experimental systems to investigate based on currently available biological knowledge, techniques in mathematical and computational analysis and methods for in vitro and in vivo experiments.

Mathematical modeling for novel cancer drug discovery and development

INTRODUCTION

Mathematical modeling enables: the in silico classification of cancers, the prediction of disease outcomes, optimization of therapy, identification of promising drug targets and prediction of resistance to anticancer drugs. In silico pre-screened drug targets can be validated by a small number of carefully selected experiments.

AREAS COVERED

This review discusses the basics of mathematical modeling in cancer drug discovery and development. The topics include in silico discovery of novel molecular drug targets, optimization of immunotherapies, personalized medicine and guiding preclinical and clinical trials. Breast cancer has been used to demonstrate the applications of mathematical modeling in cancer diagnostics, the identification of high-risk population, cancer screening strategies, prediction of tumor growth and guiding cancer treatment.

EXPERT OPINION

Mathematical models are the key components of the toolkit used in the fight against cancer. The combinatorial complexity of new drugs discovery is enormous, making systematic drug discovery, by experimentation, alone difficult if not impossible. The biggest challenges include seamless integration of growing data, information and knowledge, and making them available for a multiplicity of analyses. Mathematical models are essential for bringing cancer drug discovery into the era of Omics, Big Data and personalized medicine.

Tumor growth in complex, evolving microenvironmental geometries: a diffuse domain approach

We develop a mathematical model of tumor growth in complex, dynamic microenvironments with active, deformable membranes. Using a diffuse domain approach, the complex domain is captured implicitly using an auxiliary function and the governing equations are appropriately modified, extended and solved in a larger, regular domain. The diffuse domain method enables us to develop an efficient numerical implementation that does not depend on the space dimension or the microenvironmental geometry. We model homotypic cell-cell adhesion and heterotypic cell-basement membrane (BM) adhesion with the latter being implemented via a membrane energy that models cell-BM interactions. We incorporate simple models of elastic forces and the degradation of the BM and ECM by tumor-secreted matrix degrading enzymes. We investigate tumor progression and BM response as a function of cell-BM adhesion and the stiffness of the BM. We find tumor sizes tend to be positively correlated with cell-BM adhesion since increasing cell-BM adhesion results in thinner, more elongated tumors. Prior to invasion of the tumor into the stroma, we find a negative correlation between tumor size and BM stiffness as the elastic restoring forces tend to inhibit tumor growth. In order to model tumor invasion of the stroma, we find it necessary to downregulate cell-BM adhesiveness, which is consistent with experimental observations. A stiff BM promotes invasiveness because at early stages the opening in the BM created by MDE degradation from tumor cells tends to be narrower when the BM is stiffer. This requires invading cells to squeeze through the narrow opening and thus promotes fragmentation that then leads to enhanced growth and invasion. In three dimensions, the opening in the BM was found to increase in size even when the BM is stiff because of pressure induced by growing tumor clusters. A larger opening in the BM can increase the potential for further invasiveness by increasing the possibility that additional tumor cells could invade the stroma.

Identifying a highly-aggressive DCIS subgroup by studying intra-individual DCIS heterogeneity among invasive breast cancer patients

The heterogeneity among multiple ductal carcinoma in situ (DCIS) lesions within the same patient also diagnosed with invasive ductal carcinoma (IDC) has not been well evaluated, leaving research implications of intra-individual DCIS heterogeneity yet to be explored. In this study formalin-fixed paraffin embedded sections from 36 patients concurrently diagnosed with DCIS and IDC were evaluated by immunohistochemistry. Ten DCIS lesions from each patient were then randomly selected and scored. Our results showed that expression of PR, HER2, Ki-67, and p16 varied significantly within DCIS lesions from a single patient (P<0.05 for PR; P<1×10⁻⁸ for HER2, Ki-67 and p16). In addition, seventy-two percent of the individuals had heterogeneous expression of at least 2/6 markers. Importantly, by evaluating the expression of promising DCIS risk biomarkers (Ki-67, p53 and p16) among different DCIS subgroups classified by comparing DCIS molecular subtypes with those of adjacent normal terminal duct lobular units (TDLU) and IDC, our results suggest the existence of a highly-aggressive DCIS subgroup, which had the same molecular subtype as the adjacent IDC but not the same subtype as the adjacent normal TDLU. By using a systematic approach, our results clearly demonstrate that intra-individual heterogeneity in DCIS is very common in patients concurrently diagnosed with IDC. Our novel findings of a DCIS subpopulation with aggressive characteristics will provide a new paradigm for mechanistic studies of breast tumor progression and also have broad implications for prevention research as heterogeneous pre-invasive lesions are present in many other cancer types.

Simulating cancer growth with multiscale agent-based modeling

There have been many techniques developed in recent years to in silico model a variety of cancer behaviors. Agent-based modeling is a specific discrete-based hybrid modeling approach that allows simulating the role of diversity in cell populations as well as within each individual cell; it has therefore become a powerful modeling method widely used by computational cancer researchers. Many aspects of tumor morphology including phenotype-changing mutations, the adaptation to microenvironment, the process of angiogenesis, the influence of extracellular matrix, reactions to chemotherapy or surgical intervention, the effects of oxygen and nutrient availability, and metastasis and invasion of healthy tissues have been incorporated and investigated in agent-based models. In this review, we introduce some of the most recent agent-based models that have provided insight into the understanding of cancer growth and invasion, spanning multiple biological scales in time and space, and we further describe several experimentally testable hypotheses generated by those models. We also discuss some of the current challenges of multiscale agent-based cancer models.

Stretched cell cycle model for proliferating lymphocytes

Stochastic variation in cell cycle time is a consistent feature of otherwise similar cells within a growing population. Classic studies concluded that the bulk of the variation occurs in the G1 phase, and many mathematical models assume a constant time for traversing the S/G2/M phases. By direct observation of transgenic fluorescent fusion proteins that report the onset of S phase, we establish that dividing B and T lymphocytes spend a near-fixed proportion of total division time in S/G2/M phases, and this proportion is correlated between sibling cells. This result is inconsistent with models that assume independent times for consecutive phases. Instead, we propose a stretching model for dividing lymphocytes where all parts of the cell cycle are proportional to total division time. Data fitting based on a stretched cell cycle model can significantly improve estimates of cell cycle parameters drawn from DNA labeling data used to monitor immune cell dynamics.

The formation of tight tumor clusters affects the efficacy of cell cycle inhibitors: a hybrid model study

Cyclin-dependent kinases (CDKs) are vital in regulating cell cycle progression, and, thus, in highly proliferating tumor cells CDK inhibitors are gaining interest as potential anticancer agents. Clonogenic assay experiments are frequently used to determine drug efficacy against the survival and proliferation of cancer cells. While the anticancer mechanisms of drugs are usually described at the intracellular single-cell level, the experimental measurements are sampled from the entire cancer cell population. This approach may lead to discrepancies between the experimental observations and theoretical explanations of anticipated drug mechanisms. To determine how individual cell responses to drugs that inhibit CDKs affect the growth of cancer cell populations, we developed a spatially explicit hybrid agent-based model. In this model, each cell is equipped with internal cell cycle regulation mechanisms, but it is also able to interact physically with its neighbors. We model cell cycle progression, focusing on the G1 and G2/M cell cycle checkpoints, as well as on related essential components, such as CDK1, CDK2, cell size, and DNA damage. We present detailed studies of how the emergent properties (e.g., cluster formation) of an entire cell population depend on altered physical and physiological parameters. We analyze the effects of CDK1 and CKD2 inhibitors on population growth, time-dependent changes in cell cycle distributions, and the dynamic evolution of spatial cell patterns. We show that cell cycle inhibitors that cause cell arrest at different cell cycle phases are not necessarily synergistically super-additive. Finally, we demonstrate that the physical aspects of cell population growth, such as the formation of tight cell clusters versus dispersed colonies, alter the efficacy of cell cycle inhibitors, both in 2D and 3D simulations. This finding may have implications for interpreting the treatment efficacy results of in vitro experiments, in which treatment is applied before the cells can grow to produce clusters, especially because in vivo tumors, in contrast, form large masses before they are detected and treated.

Opportunity for selection in human health

Natural selection defined by differential survival and reproduction of individuals in populations is influenced by genetic, developmental, and environmental factors operating at every age and stage in human life history: generation of gametes, conception, birth, maturation, reproduction, senescence, and death. Biological systems are built upon a hierarchical organization nesting subcellular organelles, cells, tissues, and organs within individuals, individuals within families, and families within populations, and the latter among other populations. Natural selection often acts simultaneously at more than one level of biological organization and on specific traits, which we define as multilevel selection. Under this model, the individual is a fundamental unit of biological organization and also of selection, imbedded in a larger evolutionary context, just as it is a unit of medical intervention imbedded in larger biological, cultural, and environmental contexts. Here, we view human health and life span as necessary consequences of natural selection, operating at all levels and phases of biological hierarchy in human life history as well as in sociological and environmental milieu. An understanding of the spectrum of opportunities for natural selection will help us develop novel approaches to improving healthy life span through specific and global interventions that simultaneously focus on multiple levels of biological organization. Indeed, many opportunities exist to apply multilevel selection models employed in evolutionary biology and biodemography to improving human health at all hierarchical levels. Multilevel selection perspective provides a rational theoretical foundation for a synthesis of medicine and evolution that could lead to discovering effective predictive, preventive, palliative, potentially curative, and individualized approaches in medicine and in global health programs.

Next-generation sequencing: a powerful tool for the discovery of molecular markers in breast ductal carcinoma in situ

Mammographic screening leads to frequent biopsies and concomitant overdiagnosis of breast cancer, particularly ductal carcinoma in situ (DCIS). Some DCIS lesions rapidly progress to invasive carcinoma, whereas others remain indolent. Because we cannot yet predict which lesions will not progress, all DCIS is regarded as malignant, and many women are overtreated. Thus, there is a pressing need for a panel of molecular markers in addition to the current clinical and pathological factors to provide prognostic information. Genomic technologies such as microarrays have made major contributions to defining subtypes of breast cancer. Next-generation sequencing (NGS) modalities offer unprecedented depth of expression analysis through revealing transcriptional boundaries, mutations, rare transcripts and alternative splice variants. NGS approaches are just beginning to be applied to DCIS. Here, the authors review the applications and challenges of NGS in discovering novel potential therapeutic targets and candidate biomarkers in the premalignant progression of breast cancer.

The role of engineering approaches in analysing cancer invasion and metastasis

In the past decade, novel materials, probes and tools have enabled fundamental and applied cancer researchers to take a fresh look at the complex problem of tumour invasion and metastasis. These new tools, which include imaging modalities, controlled but complex in vitro culture conditions, and the ability to model and predict complex processes in vivo, represent an integration of traditional with novel engineering approaches; and their potential effect on quantitatively understanding tumour progression and invasion looks promising.

The Need for Integrative Computational Oncology: An Illustrated Example through MMP-Mediated Tissue Degradation

Physical oncology is a growing force in cancer research, and it is enhanced by integrative computational oncology: the fusion of novel experiments with mathematical and computational modeling. Computational models must be assessed with accurate numerical methods on correctly scaled tissues to avoid numerical artifacts that can cloud analysis. Simulation-driven analyses can only be validated by careful experiments. In this perspectives piece, we evaluate a current, widespread model of matrix metalloproteinase-driven tissue degradation during cancer invasion to illustrate that integrative computational oncology may not realize its fullest potential if either of these critical steps is neglected.

Examining the pathogenesis of breast cancer using a novel agent-based model of mammary ductal epithelium dynamics

The study of the pathogenesis of breast cancer is challenged by the long time-course of the disease process and the multi-factorial nature of generating oncogenic insults. The characterization of the longitudinal pathogenesis of malignant transformation from baseline normal breast duct epithelial dynamics may provide vital insight into the cascading systems failure that leads to breast cancer. To this end, extensive information on the baseline behavior of normal mammary epithelium and breast cancer oncogenesis was integrated into a computational model termed the Ductal Epithelium Agent-Based Model (DEABM). The DEABM is composed of computational agents that behave according to rules established from published cellular and molecular mechanisms concerning breast duct epithelial dynamics and oncogenesis. The DEABM implements DNA damage and repair, cell division, genetic inheritance and simulates the local tissue environment with hormone excretion and receptor signaling. Unrepaired DNA damage impacts the integrity of the genome within individual cells, including a set of eight representative oncogenes and tumor suppressors previously implicated in breast cancer, with subsequent consequences on successive generations of cells. The DEABM reproduced cellular population dynamics seen during the menstrual cycle and pregnancy, and demonstrated the oncogenic effect of known genetic factors associated with breast cancer, namely TP53 and Myc, in simulations spanning ∼40 years of simulated time. Simulations comparing normal to BRCA1-mutant breast tissue demonstrated rates of invasive cancer development similar to published epidemiologic data with respect to both cumulative incidence over time and estrogen-receptor status. Investigation of the modeling of ERα-positive (ER+) tumorigenesis led to a novel hypothesis implicating the transcription factor and tumor suppressor RUNX3. These data suggest that the DEABM can serve as a potentially valuable framework to augment the traditional investigatory workflow for future hypothesis generation and testing of the mechanisms of breast cancer oncogenesis.

Ovarian tumor attachment, invasion, and vascularization reflect unique microenvironments in the peritoneum: insights from xenograft and mathematical models

Ovarian cancer relapse is often characterized by metastatic spread throughout the peritoneal cavity with tumors attached to multiple organs. In this study, interaction of ovarian cancer cells with the peritoneal tumor microenvironment was evaluated in a xenograft model based on intraperitoneal injection of fluorescent SKOV3.ip1 ovarian cancer cells. Intra-vital microscopy of mixed GFP-red fluorescent protein (RFP) cell populations injected into the peritoneum demonstrated that cancer cells aggregate and attach as mixed spheroids, emphasizing the importance of homotypic adhesion in tumor formation. Electron microscopy provided high resolution structural information about local attachment sites. Experimental measurements from the mouse model were used to build a three-dimensional cellular Potts ovarian tumor model (OvTM) that examines ovarian cancer cell attachment, chemotaxis, growth, and vascularization. OvTM simulations provide insight into the relative influence of cancer cell-cell adhesion, oxygen availability, and local architecture on tumor growth and morphology. Notably, tumors on the mesentery, omentum, or spleen readily invade the "open" architecture, while tumors attached to the gut encounter barriers that restrict invasion and instead rapidly expand into the peritoneal space. Simulations suggest that rapid neovascularization of SKOV3.ip1 tumors is triggered by constitutive release of angiogenic factors in the absence of hypoxia. This research highlights the importance of cellular adhesion and tumor microenvironment in the seeding of secondary ovarian tumors on diverse organs within the peritoneal cavity. Results of the OvTM simulations indicate that invasion is strongly influenced by features underlying the mesothelial lining at different sites, but is also affected by local production of chemotactic factors. The integrated in vivo mouse model and computer simulations provide a unique platform for evaluating targeted therapies for ovarian cancer relapse.

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.

Computational medicine: translating models to clinical care

Because of the inherent complexity of coupled nonlinear biological systems, the development of computational models is necessary for achieving a quantitative understanding of their structure and function in health and disease. Statistical learning is applied to high-dimensional biomolecular data to create models that describe relationships between molecules and networks. Multiscale modeling links networks to cells, organs, and organ systems. Computational approaches are used to characterize anatomic shape and its variations in health and disease. In each case, the purposes of modeling are to capture all that we know about disease and to develop improved therapies tailored to the needs of individuals. We discuss advances in computational medicine, with specific examples in the fields of cancer, diabetes, cardiology, and neurology. Advances in translating these computational methods to the clinic are described, as well as challenges in applying models for improving patient health.

An integrated computational/experimental model of lymphoma growth

Non-Hodgkin's lymphoma is a disseminated, highly malignant cancer, with resistance to drug treatment based on molecular- and tissue-scale characteristics that are intricately linked. A critical element of molecular resistance has been traced to the loss of functionality in proteins such as the tumor suppressor p53. We investigate the tissue-scale physiologic effects of this loss by integrating in vivo and immunohistological data with computational modeling to study the spatiotemporal physical dynamics of lymphoma growth. We compare between drug-sensitive Eμ-myc Arf-/- and drug-resistant Eμ-myc p53-/- lymphoma cell tumors grown in live mice. Initial values for the model parameters are obtained in part by extracting values from the cellular-scale from whole-tumor histological staining of the tumor-infiltrated inguinal lymph node in vivo. We compare model-predicted tumor growth with that observed from intravital microscopy and macroscopic imaging in vivo, finding that the model is able to accurately predict lymphoma growth. A critical physical mechanism underlying drug-resistant phenotypes may be that the Eμ-myc p53-/- cells seem to pack more closely within the tumor than the Eμ-myc Arf-/- cells, thus possibly exacerbating diffusion gradients of oxygen, leading to cell quiescence and hence resistance to cell-cycle specific drugs. Tighter cell packing could also maintain steeper gradients of drug and lead to insufficient toxicity. The transport phenomena within the lymphoma may thus contribute in nontrivial, complex ways to the difference in drug sensitivity between Eμ-myc Arf-/- and Eμ-myc p53-/- tumors, beyond what might be solely expected from loss of functionality at the molecular scale. We conclude that computational modeling tightly integrated with experimental data gives insight into the dynamics of Non-Hodgkin's lymphoma and provides a platform to generate confirmable predictions of tumor growth.

Non-Hodgkin's lymphoma is a cancer that develops from white blood cells called lymphocytes in the immune system, whose role is to fight disease throughout the body. This cancer can spread throughout the whole body and be very lethal – in the US, one third of patients will die from this disease within five years of diagnosis. Chemotherapy is a usual treatment for lymphoma, but the cancer can become highly resistant to it. One reason is that a critical gene called p53 can become mutated and help the cancer to survive. In this work we investigate how cells with this mutation affect the cancer growth by performing experiments in mice and using a computer model. By inputting the model parameters based on data from the experiments, we are able to accurately predict the growth of the tumor as compared to tumor measurements in living mice. We conclude that computational modeling integrated with experimental data gives insight into the dynamics of Non-Hodgkin's lymphoma, and provides a platform to generate confirmable predictions of tumor growth.

Chaste: an open source C++ library for computational physiology and biology

Chaste — Cancer, Heart And Soft Tissue Environment — is an open source C++ library for the computational simulation of mathematical models developed for physiology and biology. Code development has been driven by two initial applications: cardiac electrophysiology and cancer development. A large number of cardiac electrophysiology studies have been enabled and performed, including high-performance computational investigations of defibrillation on realistic human cardiac geometries. New models for the initiation and growth of tumours have been developed. In particular, cell-based simulations have provided novel insight into the role of stem cells in the colorectal crypt. Chaste is constantly evolving and is now being applied to a far wider range of problems. The code provides modules for handling common scientific computing components, such as meshes and solvers for ordinary and partial differential equations (ODEs/PDEs). Re-use of these components avoids the need for researchers to ‘re-invent the wheel’ with each new project, accelerating the rate of progress in new applications. Chaste is developed using industrially-derived techniques, in particular test-driven development, to ensure code quality, re-use and reliability. In this article we provide examples that illustrate the types of problems Chaste can be used to solve, which can be run on a desktop computer. We highlight some scientific studies that have used or are using Chaste, and the insights they have provided. The source code, both for specific releases and the development version, is available to download under an open source Berkeley Software Distribution (BSD) licence at http://www.cs.ox.ac.uk/chaste, together with details of a mailing list and links to documentation and tutorials.

An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix

The basement membrane (BM) and extracellular matrix (ECM) play critical roles in developmental and cancer biology, and are of great interest in biomathematics. We introduce a model of mechanical cell-BM-ECM interactions that extends current (visco)elastic models (e.g. [8,16]), and connects to recent agent-based cell models (e.g. [2,3,20,26]). We model the BM as a linked series of Hookean springs, each with time-varying length, thickness, and spring constant. Each BM spring node exchanges adhesive and repulsive forces with the cell agents using potential functions. We model elastic BM-ECM interactions with analogous ECM springs. We introduce a new model of plastic BM and ECM reorganization in response to prolonged strains, and new constitutive relations that incorporate molecular-scale effects of plasticity into the spring constants. We find that varying the balance of BM and ECM elasticity alters the node spacing along cell boundaries, yielding a nonuniform BM thickness. Uneven node spacing generates stresses that are relieved by plasticity over long times. We find that elasto-viscoplastic cell shape response is critical to relieving uneven stresses in the BM. Our modeling advances and results highlight the importance of rigorously modeling of cell-BM-ECM interactions in clinically important conditions with significant membrane deformations and time-varying membrane properties, such as aneurysms and progression from in situ to invasive carcinoma.

A multiscale road map of cancer spheroids – incorporating experimental and mathematical modelling to understand cancer progression

Computational models represent a highly suitable framework, not only for testing biological hypotheses and generating new ones but also for optimising experimental strategies. As one surveys the literature devoted to cancer modelling, it is obvious that immense progress has been made in applying simulation techniques to the study of cancer biology, although the full impact has yet to be realised. For example, there are excellent models to describe cancer incidence rates or factors for early disease detection, but these predictions are unable to explain the functional and molecular changes that are associated with tumour progression. In addition, it is crucial that interactions between mechanical effects, and intracellular and intercellular signalling are incorporated in order to understand cancer growth, its interaction with the extracellular microenvironment and invasion of secondary sites. There is a compelling need to tailor new, physiologically relevant in silico models that are specialised for particular types of cancer, such as ovarian cancer owing to its unique route of metastasis, which are capable of investigating anti-cancer therapies, and generating both qualitative and quantitative predictions. This Commentary will focus on how computational simulation approaches can advance our understanding of ovarian cancer progression and treatment, in particular, with the help of multicellular cancer spheroids, and thus, can inform biological hypothesis and experimental design.

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.

Automated reconstruction algorithm for identification of 3D architectures of cribriform ductal carcinoma in situ

Ductal carcinoma in situ (DCIS) is a pre-invasive carcinoma of the breast that exhibits several distinct morphologies but the link between morphology and patient outcome is not clear. We hypothesize that different mechanisms of growth may still result in similar 2D morphologies, which may look different in 3D. To elucidate the connection between growth and 3D morphology, we reconstruct the 3D architecture of cribriform DCIS from resected patient material. We produce a fully automated algorithm that aligns, segments, and reconstructs 3D architectures from microscopy images of 2D serial sections from human specimens. The alignment algorithm is based on normalized cross correlation, the segmentation algorithm uses histogram equilization, Otsu's thresholding, and morphology techniques to segment the duct and cribra. The reconstruction method combines these images in 3D. We show that two distinct 3D architectures are indeed found in samples whose 2D histological sections are similarly identified as cribriform DCIS. These differences in architecture support the hypothesis that luminal spaces may form due to different mechanisms, either isolated cell death or merging fronds, leading to the different architectures. We find that out of 15 samples, 6 were found to have 'bubble-like' cribra, 6 were found to have 'tube-like' criba and 3 were 'unknown.' We propose that the 3D architectures found, 'bubbles' and 'tubes', account for some of the heterogeneity of the disease and may be prognostic indicators of different patient outcomes.

Modelling the effects of cell-cycle heterogeneity on the response of a solid tumour to chemotherapy: biological insights from a hybrid multiscale cellular automaton model

The therapeutic control of a solid tumour depends critically on the responses of the individual cells that constitute the entire tumour mass. A particular cell's spatial location within the tumour and intracellular interactions, including the evolution of the cell-cycle within each cell, has an impact on their decision to grow and divide. They are also influenced by external signals from other cells as well as oxygen and nutrient concentrations. Hence, it is important to take these into account when modelling tumour growth and the response to various treatment regimes ('cell-kill therapies'), including chemotherapy. In order to address this multiscale nature of solid tumour growth and its response to treatment, we propose a hybrid, individual-based approach that analyses spatio-temporal dynamics at the level of cells, linking individual cell behaviour with the macroscopic behaviour of cell organisation and the microenvironment. The individual tumour cells are modelled by using a cellular automaton (CA) approach, where each cell has its own internal cell-cycle, modelled using a system of ODEs. The internal cell-cycle dynamics determine the growth strategy in the CA model, making it more predictive and biologically relevant. It also helps to classify the cells according to their cell-cycle states and to analyse the effect of various cell-cycle dependent cytotoxic drugs. Moreover, we have incorporated the evolution of oxygen dynamics within this hybrid model in order to study the effects of the microenvironment in cell-cycle regulation and tumour treatments. An important factor from the treatment point of view is that the low concentration of oxygen can result in a hypoxia-induced quiescence (G0/G1 arrest) of the cancer cells, making them resistant to key cytotoxic drugs. Using this multiscale model, we investigate the impact of oxygen heterogeneity on the spatio-temporal patterning of the cell distribution and their cell-cycle status. We demonstrate that oxygen transport limitations result in significant heterogeneity in HIF-1 α signalling and cell-cycle status, and when these are combined with drug transport limitations, the efficacy of the therapy is significantly impaired.

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.

Integrative physical oncology

Cancer is arguably the ultimate complex biological system. Solid tumors are microstructured soft matter that evolves as a consequence of spatio-temporal events at the intracellular (e.g., signaling pathways, macromolecular trafficking), intercellular (e.g., cell-cell adhesion/communication), and tissue (e.g., cell-extracellular matrix interactions, mechanical forces) scales. To gain insight, tumor and developmental biologists have gathered a wealth of molecular, cellular, and genetic data, including immunohistochemical measurements of cell type-specific division and death rates, lineage tracing, and gain-of-function/loss-of-function mutational analyses. These data are empirically extrapolated to a diagnosis/prognosis of tissue-scale behavior, e.g., for clinical decision. Integrative physical oncology (IPO) is the science that develops physically consistent mathematical approaches to address the significant challenge of bridging the nano (nm)-micro (µm) to macro (mm, cm) scales with respect to tumor development and progression. In the current literature, such approaches are referred to as multiscale modeling. In the present article, we attempt to assess recent modeling approaches on each separate scale and critically evaluate the current 'hybrid-multiscale' models used to investigate tumor growth in the context of brain and breast cancers. Finally, we provide our perspective on the further development and the impact of IPO.