Morphology transitions induced by chemotherapy in carcinomas in situ

A Review of Mathematical and Computational Methods in Cancer Dynamics

Cancers are complex adaptive diseases regulated by the nonlinear feedback systems between genetic instabilities, environmental signals, cellular protein flows, and gene regulatory networks. Understanding the cybernetics of cancer requires the integration of information dynamics across multidimensional spatiotemporal scales, including genetic, transcriptional, metabolic, proteomic, epigenetic, and multi-cellular networks. However, the time-series analysis of these complex networks remains vastly absent in cancer research. With longitudinal screening and time-series analysis of cellular dynamics, universally observed causal patterns pertaining to dynamical systems, may self-organize in the signaling or gene expression state-space of cancer triggering processes. A class of these patterns, strange attractors, may be mathematical biomarkers of cancer progression. The emergence of intracellular chaos and chaotic cell population dynamics remains a new paradigm in systems medicine. As such, chaotic and complex dynamics are discussed as mathematical hallmarks of cancer cell fate dynamics herein. Given the assumption that time-resolved single-cell datasets are made available, a survey of interdisciplinary tools and algorithms from complexity theory, are hereby reviewed to investigate critical phenomena and chaotic dynamics in cancer ecosystems. To conclude, the perspective cultivates an intuition for computational systems oncology in terms of nonlinear dynamics, information theory, inverse problems, and complexity. We highlight the limitations we see in the area of statistical machine learning but the opportunity at combining it with the symbolic computational power offered by the mathematical tools explored.

Dynamics of the cell-mediated immune response to tumour growth

Using a hybrid cellular automaton, we investigate the transient and asymptotic dynamics of the cell-mediated immune response to tumour growth. We analyse the correspondence between this dynamics and the three phases of the theory of immunoedition: elimination, equilibrium and escape. Our results demonstrate that the immune system can keep a tumour dormant for long periods of time, but that this dormancy is based on a frail equilibrium between the mechanisms that spur the immune response and the growth of the tumour. Thus, we question the capacity of the cell-mediated immune response to sustain long periods of dormancy, as those appearing in recurrent disease. We suggest that its role might be rather to synergize with other types of tumour dormancy.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Simulation of avascular tumor growth by agent-based game model involving phenotype-phenotype interactions

All tumors, both benign and metastatic, undergo an avascular growth stage with nutrients supplied by the surrounding tissue. This avascular growth process is much easier to carry out in more qualitative and quantitative experiments starting from tumor spheroids in vitro with reliable reproducibility. Essentially, this tumor progression would be described as a sequence of phenotypes. Using agent-based simulation in a two-dimensional spatial lattice, we constructed a composite growth model in which the phenotypic behavior of tumor cells depends on not only the local nutrient concentration and cell count but also the game among cells. Our simulation results demonstrated that in silico tumors are qualitatively similar to those observed in tumor spheroid experiments. We also found that the payoffs in the game between two living cell phenotypes can influence the growth velocity and surface roughness of tumors at the same time. Finally, this current model is flexible and can be easily extended to discuss other situations, such as environmental heterogeneity and mutation.

A validated mathematical model of tumor growth including tumor-host interaction, cell-mediated immune response and chemotherapy

We consider a dynamical model of cancer growth including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. The tumor-immune and the tumor-host interactions are characterized to reproduce experimental results. A thorough dynamical analysis of the model is carried out, showing its capability to explain theoretical and empirical knowledge about tumor development. A chemotherapy treatment reproducing different experiments is also introduced. We believe that this simple model can serve as a foundation for the development of more complicated and specific cancer models.

Shape in migration: quantitative image analysis of migrating chemoresistant HCT-8 colon cancer cells

Unsuccessful cytotoxic anticancer treatments may contribute to tumor morphologic instability and consequent tissue invasion, promoting the selection of a more malignant phenotype. Indeed, morphological changes have been demonstrated to be more pronounced in strongly vs. weakly metastatic cells. By means of normalized bending energy, we have previously quantitatively defined the link between cell shape modifications and the acquisition of a more malignant phenotype by 5-FU-resistant colon cancer cells (HCT-8FUres). Such changes were significantly correlated with an increase in motility speed. Herein, we propose a method to quantitatively analyze the shape of wild and chemoresistant HCT-8 migration front cells during wound healing assay. We evaluated the reliability of parameters (area/perimeter ratio [A/p], circularity, roundness, fractal dimension, and solidity) in describing the biological behavior of the two cell lines, enabling hence in distinguishing the chemoresistant line from the other one. We found solidity index the parameter that better described the difference between chemoresistant and wild cells. Moreover, solidity is able to capture the differences between chemoresistant and wild cells at each time point of the migration process. Indeed, motility speed was found to be inversely correlated with solidity, a quantitative index of cell deformability. Deformability is an outstanding hallmark of the process leading to metastatic spread; consequently, solidity may be considered a marker of acquired metastatic property.

Optimization of vascular-targeting drugs in a computational model of tumor growth

A biophysical tool is introduced that seeks to provide a theoretical basis for helping drug design teams assess the most promising drug targets and design optimal treatment strategies. The tool is grounded in a previously validated computational model of the feedback that occurs between a growing tumor and the evolving vasculature. In this paper, the model is particularly used to explore the therapeutic effectiveness of two drugs that target the tumor vasculature: angiogenesis inhibitors (AIs) and vascular disrupting agents (VDAs). Using sensitivity analyses, the impact of VDA dosing parameters is explored, as is the effects of administering a VDA with an AI. Further, a stochastic optimization scheme is utilized to identify an optimal dosing schedule for treatment with an AI and a chemotherapeutic. The treatment regimen identified can successfully halt simulated tumor growth, even after the cessation of therapy.

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.

Quantitative shape analysis of chemoresistant colon cancer cells: correlation between morphotype and phenotype

Morphological, qualitative observations allow pathologists to correlate the shape the cells acquire with the progressive, underlying neoplastic transformation they are experienced. Cell morphology, indeed, roughly scales with malignancy. A quantitative parameter for characterizing complex irregular structures is the Normalized Bending Energy (NBE). NBE provides a global feature for shape characterization correspondent to the amount of energy needed to transform the specific shape under analysis into its lowest energy state. We hypothesized that a chemotherapy resistant cancer cell line would experience a significant change in its shape, and that such a modification might be quantified by means of NBE parameterization. We checked out the usefulness of a mathematical algorithm to distinguish wild and 5-fluorouracil (5-FU)-resistant colon cancer HCT-8 cells (HCT-8FUres). NBE values, as well as cellular and molecular parameters, were recorded in both cell populations. Results demonstrated that acquisition of drug resistance is accompanied by statistically significant morphological changes in cell membrane, as well as in biological parameters. Namely, NBE increased progressively meanwhile cells become more resistant to increasing 5-FU concentrations. These data indicate how tight the relationships between morphology and phenotype is, and they support the idea to follow a cell transition toward a drug-resistant phenotype by means of morphological monitoring.

In silico experimental modeling of cancer treatment

In silico experimental modeling of cancer involves combining findings from biological literature with computer-based models of biological systems in order to conduct investigations of hypotheses entirely in the computer laboratory. In this paper, we discuss the use of in silico modeling as a precursor to traditional clinical and laboratory research, allowing researchers to refine their experimental programs with an aim to reducing costs and increasing research efficiency. We explain the methodology of in silico experimental trials before providing an example of in silico modeling from the biomathematical literature with a view to promoting more widespread use and understanding of this research strategy.

Combining cellular automata and Lattice Boltzmann method to model multiscale avascular tumor growth coupled with nutrient diffusion and immune competition

In the last decades the Lattice Boltzmann method (LB) has been successfully used to simulate a variety of processes. The LB model describes the microscopic processes occurring at the cellular level and the macroscopic processes occurring at the continuum level with a unique function, the probability distribution function. Recently, it has been tried to couple deterministic approaches with probabilistic cellular automata (probabilistic CA) methods with the aim to model temporal evolution of tumor growths and three dimensional spatial evolution, obtaining hybrid methodologies. Despite the good results attained by CA-PDE methods, there is one important issue which has not been completely solved: the intrinsic stochastic nature of the interactions at the interface between cellular (microscopic) and continuum (macroscopic) level. CA methods are able to cope with the stochastic phenomena because of their probabilistic nature, while PDE methods are fully deterministic. Even if the coupling is mathematically correct, there could be important statistical effects that could be missed by the PDE approach. For such a reason, to be able to develop and manage a model that takes into account all these three level of complexity (cellular, molecular and continuum), we believe that PDE should be replaced with a statistic and stochastic model based on the numerical discretization of the Boltzmann equation: The Lattice Boltzmann (LB) method. In this work we introduce a new hybrid method to simulate tumor growth and immune system, by applying Cellular Automata Lattice Boltzmann (CA-LB) approach.

Fractal analysis in a systems biology approach to cancer

Cancer is a highly complex disease due to the disruption of tissue architecture. Thus, tissues, and not individual cells, are the proper level of observation for the study of carcinogenesis. This paradigm shift from a reductionist approach to a systems biology approach is long overdue. Indeed, cell phenotypes are emergent modes arising through collective non-linear interactions among different cellular and microenvironmental components, generally described by "phase space diagrams", where stable states (attractors) are embedded into a landscape model. Within this framework, cell states and cell transitions are generally conceived as mainly specified by gene-regulatory networks. However, the system's dynamics is not reducible to the integrated functioning of the genome-proteome network alone; the epithelia-stroma interacting system must be taken into consideration in order to give a more comprehensive picture. Given that cell shape represents the spatial geometric configuration acquired as a result of the integrated set of cellular and environmental cues, we posit that fractal-shape parameters represent "omics" descriptors of the epithelium-stroma system. Within this framework, function appears to follow form, and not the other way around.

Metabolism and cell shape in cancer: a fractal analysis

Fractal analysis in cancer cell investigation provided meaningful insights into the relationship between morphology and phenotype. Some reports demonstrated that changes in cell shape precede and trigger dramatic modifications in both gene expression and enzymatic function. Nonetheless, metabolomic pattern in cells undergoing shape changes have been not still reported. Our study was aimed to investigate if modifications in cancer cell morphology are associated to relevant transition in tumour metabolome, analyzed by nuclear magnetic resonance spectroscopy and principal component analysis. MCF-7 and MDA-MB-231 breast cancer cells, exposed to an experimental morphogenetic field, undergo a dramatic change in their membrane profiles. Both cell lines recover a more rounded shape, loosing spindle and invasive protrusions, acquiring a quite "normal" morphology. This result, quantified by fractal analysis, shows that normalized bending energy (a global shape characterization expressing the amount of energy needed to transform a specific shape into its lowest energy state) decreases after 48 h. Later on, a significant shift from a high to a low glycolytic phenotype was observed on both cell lines: glucose flux begins to drop off at 48 h, leading to reduced lactate accumulation, and fatty acids and citrate synthesis slow-down after 72 h. Moreover, de novo lipidogenesis is inhibited and nucleotide synthesis is reduced, as indicated by the positive correlation between glucose and formate. In conclusion, these data indicate that the reorganization of cell membrane architecture, induced by environmental cues, is followed by a relevant transition of the tumour metabolome, suggesting cells undergo a dramatic phenotypic reversion.

A 2D mechanistic model of breast ductal carcinoma in situ (DCIS) morphology and progression

Ductal carcinoma in situ (DCIS) of the breast is a non-invasive tumor in which cells proliferate abnormally, but remain confined within a duct. Although four distinguishable DCIS morphologies are recognized, the mechanisms that generate these different morphological classes remain unclear, and consequently the prognostic strength of DCIS classification is not strong. To improve the understanding of the relation between morphology and time course, we have developed a 2D in silico particle model of the growth of DCIS within a single breast duct. This model considers mechanical effects such as cellular adhesion and intra-ductal pressure, and biological features including proliferation, apoptosis, necrosis, and cell polarity. Using this model, we find that different regions of parameter space generate distinct morphological subtypes of DCIS, so elucidating the relation between morphology and time course. Furthermore, we find that tumors with similar architectures may in fact be produced through different mechanisms, and we propose future work to further disentangle the mechanisms involved in DCIS progression.

A theoretical quantitative model for evolution of cancer chemotherapy resistance

Background

Disseminated cancer remains a nearly uniformly fatal disease. While a number of effective chemotherapies are available, tumors inevitably evolve resistance to these drugs ultimately resulting in treatment failure and cancer progression. Causes for chemotherapy failure in cancer treatment reside in multiple levels: poor vascularization, hypoxia, intratumoral high interstitial fluid pressure, and phenotypic resistance to drug-induced toxicity through upregulated xenobiotic metabolism or DNA repair mechanisms and silencing of apoptotic pathways. We propose that in order to understand the evolutionary dynamics that allow tumors to develop chemoresistance, a comprehensive quantitative model must be used to describe the interactions of cell resistance mechanisms and tumor microenvironment during chemotherapy.

Ultimately, the purpose of this model is to identify the best strategies to treat different types of tumor (tumor microenvironment, genetic/phenotypic tumor heterogeneity, tumor growth rate, etc.). We predict that the most promising strategies are those that are both cytotoxic and apply a selective pressure for a phenotype that is less fit than that of the original cancer population. This strategy, known as double bind, is different from the selection process imposed by standard chemotherapy, which tends to produce a resistant population that simply upregulates xenobiotic metabolism. In order to achieve this goal we propose to simulate different tumor progression and therapy strategies (chemotherapy and glucose restriction) targeting stabilization of tumor size and minimization of chemoresistance.

Results

This work confirms the prediction of previous mathematical models and simulations that suggested that administration of chemotherapy with the goal of tumor stabilization instead of eradication would yield better results (longer subject survival) than the use of maximum tolerated doses. Our simulations also indicate that the simultaneous administration of chemotherapy and 2-deoxy-glucose does not optimize treatment outcome because when simultaneously administered these drugs are antagonists. The best results were obtained when 2-deoxy-glucose was followed by chemotherapy in two separate doses.

Conclusions

These results suggest that the maximum potential of a combined therapy may depend on how each of the drugs modifies the evolutionary landscape and that a rational use of these properties may prevent or at least delay relapse.

Reviewers

This article was reviewed by Dr Marek Kimmel and Dr Mark Little.

ImmunoGrid, an integrative environment for large-scale simulation of the immune system for vaccine discovery, design and optimization

Vaccine research is a combinatorial science requiring computational analysis of vaccine components, formulations and optimization. We have developed a framework that combines computational tools for the study of immune function and vaccine development. This framework, named ImmunoGrid combines conceptual models of the immune system, models of antigen processing and presentation, system-level models of the immune system, Grid computing, and database technology to facilitate discovery, formulation and optimization of vaccines. ImmunoGrid modules share common conceptual models and ontologies. The ImmunoGrid portal offers access to educational simulators where previously defined cases can be displayed, and to research simulators that allow the development of new, or tuning of existing, computational models. The portal is accessible at <igrid-ext.cryst.bbk.ac.uk/immunogrid>.

A multiscale mathematical model for oncolytic virotherapy

One of the most promising strategies to treat cancer is attacking it with viruses. Oncolytic viruses can kill tumor cells specifically or induce anticancer immune response. A multiscale model for virotherapy of cancer is investigated through simulations. It was found that, for intratumoral virus administration, a solid tumor can be completely eradicated or keep growing after a transient remission. Furthermore, the model reveals undamped oscillatory dynamics of tumor cells and virus populations, which demands new in vivo and in vitro quantitative experiments aiming to detect this oscillatory response. The conditions for which each one of the different tumor responses dominates, as well as the occurrence probabilities for the other nondominant therapeutic outcomes, were determined. From a clinical point of view, our findings indicate that a successful, single agent virotherapy requires a strong inhibition of the host immune response and the use of potent virus species with a high intratumoral mobility. Moreover, due to the discrete and stochastic nature of cells and their responses, an optimal range for viral cytotoxicity is predicted because the virotherapy fails if the oncolytic virus demands either a too short or a very large time to kill the tumor cell. This result suggests that the search for viruses able to destroy tumor cells very fast does not necessarily lead to a more effective control of tumor growth.

A cellular automata model of tumor-immune system interactions

We present a hybrid cellular automata-partial differential equation model of moderate complexity to describe the interactions between a growing tumor next to a nutrient source and the immune system of the host organism. The model allows both temporal and two-dimensional spatial evolution of the system under investigation and is comprised of biological cell metabolism rules derived from both the experimental and mathematical modeling literature. We present numerical simulations that display behaviors which are qualitatively similar to those exhibited in tumor-immune system interaction experiments. These include spherical tumor growth, stable and unstable oscillatory tumor growth, satellitosis and tumor infiltration by immune cells. Finally, the relationship between these different growth regimes and key system parameters is discussed.