Cell adhesion heterogeneity reinforces tumour cell dissemination: novel insights from a mathematical model

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.

Spatial heterogeneity in tumor adhesion qualifies collective cell invasion

Collective cell invasion (CCI), a canon of most invasive solid tumors, is an emergent property of the interactions between cancer cells and their surrounding extracellular matrix (ECM). However, tumor populations invariably consist of cells expressing variable levels of adhesive proteins that mediate such interactions, disallowing an intuitive understanding of how tumor invasiveness at a multicellular scale is influenced by spatial heterogeneity of cell-cell and cell-ECM adhesion. Here, we have used a Cellular Potts model-based multiscale computational framework that is constructed on the histopathological principles of glandular cancers. In earlier efforts on homogenous cancer cell populations, this framework revealed the relative ranges of interactions, including cell-cell and cell-ECM adhesion that drove collective, dispersed, and mixed multimodal invasion. Here, we constitute a tumor core of two separate cell subsets showing distinct intra- and inter-subset cell-cell or cell-ECM adhesion strengths. These two subsets of cells are arranged to varying extents of spatial intermingling, which we call the heterogeneity index (HI). We observe that low and high inter-subset cell adhesion favors invasion of high-HI and low-HI intermingled populations with distinct intra-subset cell-cell adhesion strengths, respectively. In addition, for explored values of cell-ECM adhesion strengths, populations with high HI values collectively invade better than those with lower HI values. We then asked how spatial invasion is regulated by progressively intermingled cellular subsets that are epithelial, i.e., showed high cell-cell but poor cell-ECM adhesion, and mesenchymal, i.e., with reversed adhesion strengths to the former. Here too, inter-subset adhesion plays an important role in contextualizing the proportionate relationship between HI and invasion. An exception to this relationship is seen for cases of heterogeneous cell-ECM adhesion where sub-maximal HI patterns with higher outer localization of cells with stronger ECM adhesion collectively invade better than their relatively higher-HI counterparts. Our simulations also reveal how adhesion heterogeneity qualifies collective invasion, when either cell-cell or cell-ECM adhesion type is varied but results in an invasive dispersion when both adhesion types are simultaneously altered.

Cancer cell sedimentation in 3D cultures reveals active migration regulated by self-generated gradients and adhesion sites

Cell sedimentation in 3D hydrogel cultures refers to the vertical migration of cells towards the bottom of the space. Understanding this poorly examined phenomenon may allow us to design better protocols to prevent it, as well as provide insights into the mechanobiology of cancer development. We conducted a multiscale experimental and mathematical examination of 3D cancer growth in triple negative breast cancer cells. Migration was examined in the presence and absence of Paclitaxel, in high and low adhesion environments and in the presence of fibroblasts. The observed behaviour was modeled by hypothesizing active migration due to self-generated chemotactic gradients. Our results did not reject this hypothesis, whereby migration was likely to be regulated by the MAPK and TGF-β pathways. The mathematical model enabled us to describe the experimental data in absence (normalized error<40%) and presence of Paclitaxel (normalized error<10%), suggesting inhibition of random motion and advection in the latter case. Inhibition of sedimentation in low adhesion and co-culture experiments further supported the conclusion that cells actively migrated downwards due to the presence of signals produced by cells already attached to the adhesive glass surface.

Metastasis Models: Thermodynamics and Complexity

The thermodynamic formalism of nonequilibrium systems together with the theory of complex systems and systems biology offer an appropriate theoretical framework to explain the complexity observed at the macroscopic level in physiological phenomena. In turn, they allow the establishment of an appropriate conceptual and operational framework to address the study of phenomena such as the emergence and evolution of cancer.This chapter is organized as follows: In Subheading 1, an integrated vision of these disciplines is offered for the characterization of the emergence and evolution of cancer, seen as a nonlinear dynamic system, temporally and spatially self-organized out of thermodynamic equilibrium. The development of the various mathematical models and different techniques and approaches used in the characterization of cancer metastasis is presented in Subheading 2. Subheading 3 is devoted to the time course of cancer metastasis, with particular emphasis on the epithelial-mesenchymal transition (EMT henceforth) as well as chronotherapeutic treatments. In Subheading 4, models of the spatial evolution of cancer metastasis are presented. Finally, in Subheading 5, some conclusions and remarks are presented.

Biophysical phenotype mixtures reveal advantages for tumor muscle invasion in vivo

Bladder, colon, gastric, prostate, and uterine cancers originate in organs surrounded by laminin-coated smooth muscle. In human prostate cancer, tumors that are organ confined, without extracapsular extension through muscle, have an overall cancer survival rate of up to 97% compared with 32% for metastatic disease. Our previous work modeling extracapsular extension reported the blocking of tumor invasion by mutation of a laminin-binding integrin called α6β1. Expression of the α6AA mutant resulted in a biophysical switch from cell-ECM (extracellular matrix) to cell-cell adhesion with drug sensitivity properties and an inability to invade muscle. Here we used different admixtures of α6AA and α6WT cells to test the cell heterogeneity requirements for muscle invasion. Time-lapse video microscopy revealed that tumor mixtures self-assembled into invasive networks in vitro, whereas α6AA cells assembled only as cohesive clusters. Invasion of α6AA cells into and through live muscle occurred using a 1:1 mixture of α6AA and α6WT cells. Electric cell-substrate impedance sensing measurements revealed that compared with α6AA cells, invasion-competent α6WT cells were 2.5-fold faster at closing a cell-ECM or cell-cell wound, respectively. Cell-ECM rebuilding kinetics show that an increased response occurred in mixtures since the response was eightfold greater compared with populations containing only one cell type. A synthetic cell adhesion cyclic peptide called MTI-101 completely blocked electric cell-substrate impedance sensing cell-ECM wound recovery that persisted in vitro up to 20 h after the wound. Treatment of tumor-bearing animals with 10 mg/kg MTI-101 weekly resulted in a fourfold decrease of muscle invasion by tumor and a decrease of the depth of invasion into muscle comparable to the α6AA cells. Taken together, these data suggest that mixed biophysical phenotypes of tumor cells within a population can provide functional advantages for tumor invasion into and through muscle that can be potentially inhibited by a synthetic cell adhesion molecule.

Quantification of spatial subclonal interactions enhancing the invasive phenotype of pediatric glioma

Diffuse midline gliomas (DMGs) are highly aggressive, incurable childhood brain tumors. They present a clinical challenge due to many factors, including heterogeneity and diffuse infiltration, complicating disease management. Recent studies have described the existence of subclonal populations that may co-operate to drive pro-tumorigenic processes such as cellular invasion. However, a precise quantification of subclonal interactions is lacking, a problem that extends to other cancers. In this study, we combine spatial computational modeling of cellular interactions during invasion with co-evolution experiments of clonally disassembled patient-derived DMG cells. We design a Bayesian inference framework to quantify spatial subclonal interactions between molecular and phenotypically distinct lineages with different patterns of invasion. We show how this approach could discriminate genuine interactions, where one clone enhanced the invasive phenotype of another, from those apparently only due to the complex dynamics of spatially restricted growth. This study provides a framework for the quantification of subclonal interactions in DMG.

Comprehensive Analyses of Glucose Metabolism in Glioma Reveal the Glioma-Promoting Effect of GALM

Glioma is the most common tumor with the worst prognosis in the central nervous system. Current studies showed that glucose metabolism could affect the malignant progression of tumors. However, the study on the dysregulation of glucose metabolism in glioma is still limited. Herein, we firstly screened 48 differentially expressed glucose metabolism-related genes (DE-GMGs) by comparing glioblastomas to low-grade gliomas. Then a glucose metabolism-related gene (GMG)-based model (PC, lactate dehydrogenase A (LDHA), glucuronidase beta (GUSB), galactosidase beta 1 (GLB1), galactose mutarotase (GALM), or fructose-bisphosphatase 1 (FBP1)) was constructed by a protein-protein interaction (PPI) network and Lasso regression. Thereinto, the high-risk group encountered a worse prognosis than the low-risk group, and the M2 macrophage was positively relevant to the risk score. Various classical tumor-related functions were enriched by Gene Ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) analyses. Since protein GALM was rarely studied in glioma, we detected high expression of GALM by western blot and immunohistochemistry in glioma tissues. And experiments in vitro showed that GALM could promote the epithelial-to-mesenchymal transition (EMT) process of glioma cells and could be regulated by TNFAIP3 in glioma cells. Overall, our study revealed the critical role of glucose metabolism in the prognosis of patients with glioma. Furthermore, we demonstrated that GALM was significantly related to the malignancy of glioma and could promote glioma cells' EMT process.

BIO-LGCA: A cellular automaton modelling class for analysing collective cell migration

Collective dynamics in multicellular systems such as biological organs and tissues plays a key role in biological development, regeneration, and pathological conditions. Collective tissue dynamics—understood as population behaviour arising from the interplay of the constituting discrete cells—can be studied with on- and off-lattice agent-based models. However, classical on-lattice agent-based models, also known as cellular automata, fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. To overcome drawbacks of classical on-lattice models, we introduce an on-lattice, agent-based modelling class for collective cell migration, which we call biological lattice-gas cellular automaton (BIO-LGCA). The BIO-LGCA is characterised by synchronous time updates, and the explicit consideration of individual cell velocities. While rules in classical cellular automata are typically chosen ad hoc, rules for cell-cell and cell-environment interactions in the BIO-LGCA can also be derived from experimental cell migration data or biophysical laws for individual cell migration. We introduce elementary BIO-LGCA models of fundamental cell interactions, which may be combined in a modular fashion to model complex multicellular phenomena. We exemplify the mathematical mean-field analysis of specific BIO-LGCA models, which allows to explain collective behaviour. The first example predicts the formation of clusters in adhesively interacting cells. The second example is based on a novel BIO-LGCA combining adhesive interactions and alignment. For this model, our analysis clarifies the nature of the recently discovered invasion plasticity of breast cancer cells in heterogeneous environments.

Pattern formation during embryonic development and pathological tissue dynamics, such as cancer invasion, emerge from individual intercellular interactions. In order to study the impact of single cell dynamics and cell-cell interactions on tissue behaviour, one needs to develop space-time-dependent on- or off-lattice agent-based models (ABMs), which consider the behaviour of individual cells. However, classical on-lattice agent-based models also known as cellular automata fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. Here, we present the rule- and lattice-based BIO-LGCA modelling class which allows for (i) rigorous derivation of rules from biophysical laws and/or experimental data, (ii) mathematical analysis of collective migration, and (iii) computationally efficient simulations.

Matrix adhesion and remodeling diversifies modes of cancer invasion across spatial scales

The metastasis of malignant epithelial tumors begins with the egress of transformed cells from the confines of their basement membrane (BM) to their surrounding collagen-rich stroma. Invasion can be morphologically diverse: when breast cancer cells are separately cultured within BM-like matrix, collagen I (Coll I), or a combination of both, they exhibit collective-, dispersed mesenchymal-, and a mixed collective-dispersed (multimodal)- invasion, respectively. In this paper, we asked how distinct these invasive modes are with respect to the cellular and microenvironmental cues that drive them. A rigorous computational exploration of invasion was performed within an experimentally motivated Cellular Potts-based modeling environment. The model comprised of adhesive interactions between cancer cells, BM- and Coll I-like extracellular matrix (ECM), and reaction-diffusion-based remodeling of ECM. The model outputs were parameters cognate to dispersed- and collective- invasion. A clustering analysis of the output distribution curated through a careful examination of subsumed phenotypes suggested at least four distinct invasive states: dispersed, papillary-collective, bulk-collective, and multimodal, in addition to an indolent/non-invasive state. Mapping input values to specific output clusters suggested that each of these invasive states are specified by distinct input signatures of proliferation, adhesion and ECM remodeling. In addition, specific input perturbations allowed transitions between the clusters and revealed the variation in the robustness between the invasive states. Our systems-level approach proffers quantitative insights into how the diversity in ECM microenvironments may steer invasion into diverse phenotypic modes during early dissemination of breast cancer and contributes to tumor heterogeneity.

The transcription factor Rreb1 regulates epithelial architecture, invasiveness, and vasculogenesis in early mouse embryos

Ras-responsive element-binding protein 1 (Rreb1) is a zinc-finger transcription factor acting downstream of RAS signaling. Rreb1 has been implicated in cancer and Noonan-like RASopathies. However, little is known about its role in mammalian non-disease states. Here, we show that Rreb1 is essential for mouse embryonic development. Loss of Rreb1 led to a reduction in the expression of vasculogenic factors, cardiovascular defects, and embryonic lethality. During gastrulation, the absence of Rreb1 also resulted in the upregulation of cytoskeleton-associated genes, a change in the organization of F-ACTIN and adherens junctions within the pluripotent epiblast, and perturbed epithelial architecture. Moreover, Rreb1 mutant cells ectopically exited the epiblast epithelium through the underlying basement membrane, paralleling cell behaviors observed during metastasis. Thus, disentangling the function of Rreb1 in development should shed light on its role in cancer and other diseases involving loss of epithelial integrity.

Ni-Nitrilotriacetic Acid Affinity SELEX Method for Selection of DNA Aptamers Specific to the N-Cadherin Protein

Nucleic acid aptamers are single-stranded oligonucleotides that may be evolved for affinity and specificity for their targets and can be easily produced, regenerated, and stabilized. In this study, we adapted Ni-NTA (nickle-charged nitrilotriacetic acid) affinity-chromatography in the development of single-stranded DNA aptamers against N-cadherin protein by systematic evolution of ligands by exponential enrichment (SELEX). After ten rounds of selection, two aptamers, designated NS13 and NC23, were selected, which showed low dissociation constants of 93 and 174 nM, respectively. The 5'-carboxyfluorescein-labeled NS13 was used for the sensitive detection of N-cadherin protein by the enzyme-linked oligonucleotide assay (ELONA) method.

Exosomes Enhance Adhesion and Osteogenic Differentiation of Initial Bone Marrow Stem Cells on Titanium Surfaces

Successful osseointegration involves the biological behavior of bone marrow stem cells (BMSCs) on an implant surface; however, the role of BMSC-derived extracellular vesicles (EVs)/exosomes in osseointegration is little known. This study aimed to: (i) explore the interaction force between exosomes (Exo) and cells on a titanium surface; (ii) discuss whether the morphology and biological behavior of BMSCs are affected by exosomes; and (iii) preliminarily investigate the mechanism by which exosomes regulate cells on Ti surface. Exosomes secreted by rat BMSCs were collected by ultracentrifugation and analyzed using transmission electron microscopy and nanoparticle tracking analysis. Confocal fluorescence microscopy, scanning electron microscopy, Cell Counting Kit-8 (CCK-8), quantitative real-time polymerase chain reaction techniques, and alkaline phosphatase bioactivity, Alizarin Red staining, and quantification were used to investigate the exosomes that adhere to the Ti plates under different treatments as well as the morphological change, adhesion, spread, and differentiation of BMSCs. We found that exosomes were efficiently internalized and could regulate cell morphology and promoted the adhesion, spreading, and osteogenic differentiation of BMSCs. These were achieved partly by activating the RhoA/ROCK signaling pathway. Our discovery presents a new insight into the positive regulatory effect of exosomes on the biological behaviors of BMSCs on Ti surface and provides a novel route to modify the surface of a Ti implant.

Mini-review: advances in 3D bioprinting of vascularized constructs

3D in vitro constructs have gained more and more relevance in tissue engineering and in cancer-modeling. In recent years, with the development of thicker and more physiologically relevant tissue patches, the integration of a vascular network has become pivotal, both for sustaining the construct in vitro and to help the integration with the host tissue once implanted. Since 3D bioprinting is rising to be one of the most versatile methods to create vascularized constructs, we here briefly review the most promising advances in bioprinting techniques.

Multi-scale analysis and modelling of collective migration in biological systems

Collective migration has become a paradigm for emergent behaviour in systems of moving and interacting individual units resulting in coherent motion. In biology, these units are cells or organisms. Collective cell migration is important in embryonic development, where it underlies tissue and organ formation, as well as pathological processes, such as cancer invasion and metastasis. In animal groups, collective movements may enhance individuals' decisions and facilitate navigation through complex environments and access to food resources. Mathematical models can extract unifying principles behind the diverse manifestations of collective migration. In biology, with a few exceptions, collective migration typically occurs at a 'mesoscopic scale' where the number of units ranges from only a few dozen to a few thousands, in contrast to the large systems treated by statistical mechanics. Recent developments in multi-scale analysis have allowed linkage of mesoscopic to micro- and macroscopic scales, and for different biological systems. The articles in this theme issue on 'Multi-scale analysis and modelling of collective migration' compile a range of mathematical modelling ideas and multi-scale methods for the analysis of collective migration. These approaches (i) uncover new unifying organization principles of collective behaviour, (ii) shed light on the transition from single to collective migration, and (iii) allow us to define similarities and differences of collective behaviour in groups of cells and organisms. As a common theme, self-organized collective migration is the result of ecological and evolutionary constraints both at the cell and organismic levels. Thereby, the rules governing physiological collective behaviours also underlie pathological processes, albeit with different upstream inputs and consequences for the group. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Modeling How Heterogeneity in Cell Cycle Length Affects Cancer Cell Growth Dynamics in Response to Treatment

Tumors are complex, dynamic, and adaptive biological systems characterized by high heterogeneity at genetic, epigenetic, phenotypic, as well as tissue microenvironmental level. In this work, utilizing cellular automata methods, we focus on intrinsic heterogeneity with respect to cell cycle duration and explore whether and to what extent this heterogeneity affects cancer cell growth dynamics when cytotoxic treatment is applied. We assume that treatment acts on cancer cells specifically during mitosis and compare it with a (cell cycle-non-specific) cytotoxic treatment that acts randomly regardless of the cell cycle phase. We simulate the spatiotemporal evolution of tumor cells with different initial spatial configurations and different cell length probability distributions. We observed that in heterogeneous populations, strong selection forces act on cancer cells favoring the faster cells, when the death rates are lower than the proliferation rates. However, at higher mitotic death rates, selection of the slower proliferative cells is favored, leading to slower post-treatment regrowth rates, as compared to untreated growth. Of note, random cell death progressively eliminates the slower proliferative cells, consistently, favoring highly proliferative phenotypes. Interestingly, compared to the monoclonal populations that exhibit complete response at high random death rates, emergent resistance arises naturally in heterogeneous populations during treatment. As divergent selection forces may act on a heterogeneous cancer cell population, we argue that treatment plan selection can considerably alter the post-treatment tumor dynamics, cell survival, and emergence of resistance, proving its significant biological and therapeutic impact.

A generalizable data-driven multicellular model of pancreatic ductal adenocarcinoma

BACKGROUND

Mechanistic models, when combined with pertinent data, can improve our knowledge regarding important molecular and cellular mechanisms found in cancer. These models make the prediction of tissue-level response to drug treatment possible, which can lead to new therapies and improved patient outcomes. Here we present a data-driven multiscale modeling framework to study molecular interactions between cancer, stromal, and immune cells found in the tumor microenvironment. We also develop methods to use molecular data available in The Cancer Genome Atlas to generate sample-specific models of cancer.

RESULTS

By combining published models of different cells relevant to pancreatic ductal adenocarcinoma (PDAC), we built an agent-based model of the multicellular pancreatic tumor microenvironment, formally describing cell type-specific molecular interactions and cytokine-mediated cell-cell communications. We used an ensemble-based modeling approach to systematically explore how variations in the tumor microenvironment affect the viability of cancer cells. The results suggest that the autocrine loop involving EGF signaling is a key interaction modulator between pancreatic cancer and stellate cells. EGF is also found to be associated with previously described subtypes of PDAC. Moreover, the model allows a systematic exploration of the effect of possible therapeutic perturbations; our simulations suggest that reducing bFGF secretion by stellate cells will have, on average, a positive impact on cancer apoptosis.

CONCLUSIONS

The developed framework allows model-driven hypotheses to be generated regarding therapeutically relevant PDAC states with potential molecular and cellular drivers indicating specific intervention strategies.

Computational models to explore the complexity of the epithelial to mesenchymal transition in cancer

Epithelial to mesenchymal transition (EMT) is a complex biological process that plays a key role in cancer progression and metastasis formation. Its activation results in epithelial cells losing adhesion and polarity and becoming capable of migrating from their site of origin. At this step the disease is generally considered incurable. As EMT execution involves several individual molecular components, connected by nontrivial relations, in vitro techniques are often inadequate to capture its complexity. Computational models can be used to complement experiments and provide additional knowledge difficult to build up in a wetlab. Indeed in silico analysis gives the user total control on the system, allowing to identify the contribution of each independent element. In the following, two kinds of approaches to the computational study of EMT will be presented. The first relies on signal transduction networks description and details how changes in gene expression could influence this process, both focusing on specific aspects of the EMT and providing a general frame for this phenomenon easily comparable with experimental data. The second integrates single cell and population level descriptions in a multiscale model that can be considered a more accurate representation of the EMT. The advantages and disadvantages of each approach will be highlighted, together with the importance of coupling computational and experimental results. Finally, the main challenges that need to be addressed to improve our knowledge of the role of EMT in the neoplastic disease and the scientific and translational value of computational models in this respect will be presented. This article is categorized under: Analytical and Computational Methods > Computational Methods.

Modeling the Human Bone-Tumor Niche: Reducing and Replacing the Need for Animal Data

Bone is the most common site for cancer metastasis. Understanding the interactions within the complex, heterogeneous bone-tumor microenvironment is essential for the development of new therapeutics. Various animal models of tumor-induced bone disease are routinely used to provide valuable information on the relationship between cancer cells and the skeleton. However, new model systems exist that offer an alternative approach to the use of animals and might more accurately reveal the cellular interactions occurring within the human bone-tumor niche. This review highlights replacement models that mimic the bone microenvironment and where cancer metastases and tumor growth might be assessed alongside bone turnover. Such culture models include the use of calcified regions of animal tissue and scaffolds made from bone mineral hydroxyapatite, synthetic polymers that can be manipulated during manufacture to create structures resembling trabecular bone surfaces, gel composites that can be modified for stiffness and porosity to resemble conditions in the tumor-bone microenvironment. Possibly the most accurate model system involves the use of fresh human bone samples, which can be cultured ex vivo in the presence of human tumor cells and demonstrate similar cancer cell-bone cell interactions as described in vivo. In addition, the use of mathematical modeling and computational biology approaches provide an alternative to preliminary animal testing. The use of such models offers the capacity to mimic significant elements of the human bone-tumor environment, and complement, refine, or replace the use of preclinical models. © 2020 The Authors. JBMR Plus published by Wiley Periodicals, Inc. on behalf of American Society for Bone and Mineral Research.

A Review of Cell-Based Computational Modeling in Cancer Biology

Cancer biology involves complex, dynamic interactions between cancer cells and their tissue microenvironments. Single-cell effects are critical drivers of clinical progression. Chemical and mechanical communication between tumor and stromal cells can co-opt normal physiologic processes to promote growth and invasion. Cancer cell heterogeneity increases cancer's ability to test strategies to adapt to microenvironmental stresses. Hypoxia and treatment can select for cancer stem cells and drive invasion and resistance. Cell-based computational models (also known as discrete models, agent-based models, or individual-based models) simulate individual cells as they interact in virtual tissues, which allows us to explore how single-cell behaviors lead to the dynamics we observe and work to control in cancer systems. In this review, we introduce the broad range of techniques available for cell-based computational modeling. The approaches can range from highly detailed models of just a few cells and their morphologies to millions of simpler cells in three-dimensional tissues. Modeling individual cells allows us to directly translate biologic observations into simulation rules. In many cases, individual cell agents include molecular-scale models. Most models also simulate the transport of oxygen, drugs, and growth factors, which allow us to link cancer development to microenvironmental conditions. We illustrate these methods with examples drawn from cancer hypoxia, angiogenesis, invasion, stem cells, and immunosurveillance. An ecosystem of interoperable cell-based simulation tools is emerging at a time when cloud computing resources make software easier to access and supercomputing resources make large-scale simulation studies possible. As the field develops, we anticipate that high-throughput simulation studies will allow us to rapidly explore the space of biologic possibilities, prescreen new therapeutic strategies, and even re-engineer tumor and stromal cells to bring cancer systems under control.

Mathematical models of tumor cell proliferation: A review of the literature

INTRODUCTION

A defining hallmark of cancer is aberrant cell proliferation. Efforts to understand the generative properties of cancer cells span all biological scales: from genetic deviations and alterations of metabolic pathways to physical stresses due to overcrowding, as well as the effects of therapeutics and the immune system. While these factors have long been studied in the laboratory, mathematical and computational techniques are being increasingly applied to help understand and forecast tumor growth and treatment response. Advantages of mathematical modeling of proliferation include the ability to simulate and predict the spatiotemporal development of tumors across multiple experimental scales. Central to proliferation modeling is the incorporation of available biological data and validation with experimental data. Areas covered: We present an overview of past and current mathematical strategies directed at understanding tumor cell proliferation. We identify areas for mathematical development as motivated by available experimental and clinical evidence, with a particular emphasis on emerging, non-invasive imaging technologies. Expert commentary: The data required to legitimize mathematical models are often difficult or (currently) impossible to obtain. We suggest areas for further investigation to establish mathematical models that more effectively utilize available data to make informed predictions on tumor cell proliferation.

Integrating in vitro experiments with in silico approaches for Glioblastoma invasion: the role of cell-to-cell adhesion heterogeneity

Glioblastoma cells adopt migration strategies to invade into the brain parenchyma ranging from individual to collective mechanisms, whose role and dynamics are not yet fully understood. In this work, we explore Glioblastoma heterogeneity and recapitulate its invasive patterns both in vitro, by utilizing primary cells along with the U87MG cell line, and in silico, by adopting discrete, individual cell-based mathematics. Glioblastoma cells are cultured three-dimensionally in an ECM-like substrate. The primary Glioblastoma spheroids adopt a novel cohesive pattern, mimicking perivascular invasion in the brain, while the U87MG adopt a typical, starburst invasive pattern under the same experimental setup. Mathematically, we focus on the role of the intrinsic heterogeneity with respect to cell-to-cell adhesion. Our proposed mathematical approach mimics the invasive morphologies observed in vitro and predicts the dynamics of tumour expansion. The role of the proliferation and migration is also explored showing that their effect on tumour morphology is different per cell type. The proposed model suggests that allowing cell-to-cell adhesive heterogeneity within the tumour population is sufficient for variable invasive morphologies to emerge which remain originally undetectable by conventional imaging, indicating that exploration in pathological samples is needed to improve our understanding and reveal potential patient-specific therapeutic targets.

The biology and mathematical modelling of glioma invasion: a review

Adult gliomas are aggressive brain tumours associated with low patient survival rates and limited life expectancy. The most important hallmark of this type of tumour is its invasive behaviour, characterized by a markedly phenotypic plasticity, infiltrative tumour morphologies and the ability of malignant progression from low- to high-grade tumour types. Indeed, the widespread infiltration of healthy brain tissue by glioma cells is largely responsible for poor prognosis and the difficulty of finding curative therapies. Meanwhile, mathematical models have been established to analyse potential mechanisms of glioma invasion. In this review, we start with a brief introduction to current biological knowledge about glioma invasion, and then critically review and highlight future challenges for mathematical models of glioma invasion.

Alternative splicing of ALCAM enables tunable regulation of cell-cell adhesion through differential proteolysis

While many adhesion receptors are known to influence tumor progression, the mechanisms by which they dynamically regulate cell-cell adhesion remain elusive. We previously identified Activated Leukocyte Cell Adhesion Molecule (ALCAM) as a clinically relevant driver of metastasis and hypothesized that a tunable mechanism of ectodomain shedding regulates its contribution to dissemination. To test this hypothesis, we examined an under-explored ALCAM splice variant (ALCAM-Iso2) and demonstrated that loss of the membrane-proximal region of ALCAM (exon 13) increased metastasis four-fold. Mechanistic studies identified a novel MMP14-dependent membrane distal cleavage site in ALCAM-Iso2, which mediated a ten-fold increase in shedding, thereby decreasing cellular cohesion. Importantly, the loss of cohesion is not limited to the cell capable of shedding because the released extracellular domain diminished cohesion of non-shedding cells through disruption of ALCAM-ALCAM interactions. ALCAM-Iso2-dominated expression in bladder cancer tissue, compared to normal bladder, further emphasizes that ALCAM alternative splicing may contribute to clinical disease progression. The requirement for both the loss of exon 13 and the gain of metalloprotease activity suggests that ALCAM shedding and concomitant regulation of tumor cell adhesion is a locally tunable process.

Cellular automaton models for time-correlated random walks: derivation and analysis

Many diffusion processes in nature and society were found to be anomalous, in the sense of being fundamentally different from conventional Brownian motion. An important example is the migration of biological cells, which exhibits non-trivial temporal decay of velocity autocorrelation functions. This means that the corresponding dynamics is characterized by memory effects that slowly decay in time. Motivated by this we construct non-Markovian lattice-gas cellular automata models for moving agents with memory. For this purpose the reorientation probabilities are derived from velocity autocorrelation functions that are given a priori; in that respect our approach is “data-driven”. Particular examples we consider are velocity correlations that decay exponentially or as power laws, where the latter functions generate anomalous diffusion. The computational efficiency of cellular automata combined with our analytical results paves the way to explore the relevance of memory and anomalous diffusion for the dynamics of interacting cell populations, like confluent cell monolayers and cell clustering.

Amoeboid-mesenchymal migration plasticity promotes invasion only in complex heterogeneous microenvironments

During tissue invasion individual tumor cells exhibit two interconvertible migration modes, namely mesenchymal and amoeboid migration. The cellular microenvironment triggers the switch between both modes, thereby allowing adaptation to dynamic conditions. It is, however, unclear if this amoeboid-mesenchymal migration plasticity contributes to a more effective tumor invasion. We address this question with a mathematical model, where the amoeboid-mesenchymal migration plasticity is regulated in response to local extracellular matrix resistance. Our numerical analysis reveals that extracellular matrix structure and presence of a chemotactic gradient are key determinants of the model behavior. Only in complex microenvironments, if the extracellular matrix is highly heterogeneous and a chemotactic gradient directs migration, the amoeboid-mesenchymal migration plasticity allows a more widespread invasion compared to the non-switching amoeboid and mesenchymal modes. Importantly, these specific conditions are characteristic for in vivo tumor invasion. Thus, our study suggests that in vitro systems aiming at unraveling the underlying molecular mechanisms of tumor invasion should take into account the complexity of the microenvironment by considering the combined effects of structural heterogeneities and chemical gradients on cell migration.