Integrative mathematical oncology

Integrated intravital microscopy and mathematical modeling to optimize nanotherapeutics delivery to tumors

Inefficient vascularization hinders the optimal transport of cell nutrients, oxygen, and drugs to cancer cells in solid tumors. Gradients of these substances maintain a heterogeneous cell-scale microenvironment through which drugs and their carriers must travel, significantly limiting optimal drug exposure. In this study, we integrate intravital microscopy with a mathematical model of cancer to evaluate the behavior of nanoparticle-based drug delivery systems designed to circumvent biophysical barriers. We simulate the effect of doxorubicin delivered via porous 1000 x 400 nm plateloid silicon particles to a solid tumor characterized by a realistic vasculature, and vary the parameters to determine how much drug per particle and how many particles need to be released within the vasculature in order to achieve remission of the tumor. We envision that this work will contribute to the development of quantitative measures of nanoparticle design and drug loading in order to optimize cancer treatment via nanotherapeutics.

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.

A Systems Biology Approach in Therapeutic Response Study for Different Dosing Regimens-a Modeling Study of Drug Effects on Tumor Growth using Hybrid Systems

Motivated by the frustration of translation of research advances in the molecular and cellular biology of cancer into treatment, this study calls for cross-disciplinary efforts and proposes a methodology of incorporating drug pharmacology information into drug therapeutic response modeling using a computational systems biology approach. The objectives are two fold. The first one is to involve effective mathematical modeling in the drug development stage to incorporate preclinical and clinical data in order to decrease costs of drug development and increase pipeline productivity, since it is extremely expensive and difficult to get the optimal compromise of dosage and schedule through empirical testing. The second objective is to provide valuable suggestions to adjust individual drug dosing regimens to improve therapeutic effects considering most anticancer agents have wide inter-individual pharmacokinetic variability and a narrow therapeutic index. A dynamic hybrid systems model is proposed to study drug antitumor effect from the perspective of tumor growth dynamics, specifically the dosing and schedule of the periodic drug intake, and a drug’s pharmacokinetics and pharmacodynamics information are linked together in the proposed model using a state-space approach. It is proved analytically that there exists an optimal drug dosage and interval administration point, and demonstrated through simulation study.

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.

Cellular modeling of cancer invasion: integration of in silico and in vitro approaches

Cancer invasion is one of the hallmarks of cancer and a prerequisite for cancer metastasis. However, the invasive process is very complex, depending on multiple correlated intrinsic and environmental factors, and thus is difficult to study experimentally in a fully controlled way. Therefore, there is an increased demand for interdisciplinary integrated approaches combining laboratory experiments with multiscale in silico modeling. In this review, we will summarize current computational techniques applicable to model cancer invasion in silico, with a special focus on a class of individual-cell-based models developed in our laboratories. We also discuss their integration with traditional and novel in vitro experimentation, including new invasion assays whose design was inspired by computational modeling.

Evolutionary mechanisms and diversity in cancer

The recently introduced genome theory of cancer evolution provides a new framework for evolutionary studies on cancer. In particular, the established relationship between the large number of individual molecular mechanisms and the general evolutionary mechanism of cancer calls upon a change in our strategies that have been based on the characterization of common cancer gene mutations and their defined pathways. To further explain the significance of the genome theory of cancer evolution, a brief review will be presented describing the various attempts to illustrate the evolutionary mechanism of cancer, followed by further analysis of some key components of somatic cell evolution, including the diversity of biological systems, the multiple levels of information systems and control systems, the two phases (the punctuated or discontinuous phase and gradual Darwinian stepwise phase) and dynamic patterns of somatic cell evolution where genome replacement is the driving force. By linking various individual molecular mechanisms to the level of genome population diversity and tumorigenicity, the general mechanism of cancer has been identified as the evolutionary mechanism of cancer, which can be summarized by the following three steps including stress-induced genome instability, population diversity or heterogeneity, and genome-mediated macroevolution. Interestingly, the evolutionary mechanism is equal to the collective aggregate of all individual molecular mechanisms. This relationship explains why most of the known molecular mechanisms can contribute to cancer yet there is no single dominant mechanism for the majority of clinical cases. Despite the fact that each molecular mechanism can serve as a system stress and initiate the evolutionary process, to achieve cancer, multiple cycles of genome-mediated macroevolution are required and are a stochastically determined event. Finally, the potential clinical implications of the evolutionary mechanism of cancer are briefly reviewed.

Unraveling the complex regulatory relationships between metabolism and signal transduction in cancer

Cancer cells exhibit an altered metabolic phenotype, known as the Warburg effect, which is characterized by high rates of glucose uptake and glycolysis, even under aerobic conditions. The Warburg effect appears to be an intrinsic component of most cancers and there is evidence linking cancer progression to mutations, translocations, and alternative splicing of genes that directly code for or have downstream effects on key metabolic enzymes. Many of the same signaling pathways are routinely dysregulated in cancer and a number of important oncogenic signaling pathways play important regulatory roles in central carbon metabolism. Unraveling the complex regulatory relationship between cancer metabolism and signaling requires the application of systems biology approaches. Here we discuss computational approaches for modeling protein signal transduction and metabolism as well as how the regulatory relationship between these two important cellular processes can be combined into hybrid models.

Integrating Imaging Data into Predictive Biomathematical and Biophysical Models of Cancer

While there is a mature literature on biomathematical and biophysical modeling in cancer, many of the existing approaches are not of clinical utility, as they require input data that are extremely difficult to obtain in an intact organism, and/or require a large number of assumptions on the free parameters included in the models. Thus, there has only been very limited application of such models to solve problems of clinical import. More recently, however, there has been increased activity at the interface of quantitative, noninvasive imaging data, and tumor mathematical modeling. In addition to reporting on bulk tumor morphology and volume, emerging imaging techniques can quantitatively report on for example tumor vascularity, glucose metabolism, cell density and proliferation, and hypoxia. In this paper, we first motivate the problem of predicting therapy response by highlighting some (acknowledged) shortcomings in existing methods. We then provide introductions to a number of representative quantitative imaging methods and describe how they are currently (and potentially can be) used to initialize and constrain patient specific mathematical and biophysical models of tumor growth and treatment response, thereby increasing the clinical utility of such approaches. We conclude by highlighting some of the exciting research directions when one integrates quantitative imaging and tumor modeling.

Statistical mechanics model of angiogenic tumor growth

We examine a lattice model of tumor growth where the survival of tumor cells depends on the supplied nutrients. When such a supply is random, the extinction of tumors belongs to the directed percolation universality class. However, when the supply is correlated with the distribution of tumor cells, which as we suggest might mimic the angiogenic growth, the extinction shows different critical behavior. Such a correlation affects also the morphology of the growing tumors and drastically raises tumor-survival probability.

Numerical simulation of a thermodynamically consistent four-species tumor growth model

In this paper, we develop a thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures. Unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models. A mixed finite element spatial discretization is developed and implemented to provide numerical results demonstrating the range of solutions this model can produce. A time-stepping algorithm is then presented for this system, which is shown to be first order accurate and energy gradient stable. The results of an array of numerical experiments are presented, which demonstrate a wide range of solutions produced by various choices of model parameters.

Pharmacokinetics and pharmacodynamics of VEGF-neutralizing antibodies

Background

Vascular endothelial growth factor (VEGF) is a potent regulator of angiogenesis, and its role in cancer biology has been widely studied. Many cancer therapies target angiogenesis, with a focus being on VEGF-mediated signaling such as antibodies to VEGF. However, it is difficult to predict the effects of VEGF-neutralizing agents. We have developed a whole-body model of VEGF kinetics and transport under pathological conditions (in the presence of breast tumor). The model includes two major VEGF isoforms VEGF₁₂₁ and VEGF₁₆₅, receptors VEGFR1, VEGFR2 and co-receptors Neuropilin-1 and Neuropilin-2. We have added receptors on parenchymal cells (muscle fibers and tumor cells), and incorporated experimental data for the cell surface density of receptors on the endothelial cells, myocytes, and tumor cells. The model is applied to investigate the action of VEGF-neutralizing agents (called "anti-VEGF") in the treatment of cancer.

Results

Through a sensitivity study, we examine how model parameters influence the level of free VEGF in the tumor, a measure of the response to VEGF-neutralizing drugs. We investigate the effects of systemic properties such as microvascular permeability and lymphatic flow, and of drug characteristics such as the clearance rate and binding affinity. We predict that increasing microvascular permeability in the tumor above 10^-5 cm/s elicits the undesired effect of increasing tumor interstitial VEGF concentration beyond even the baseline level. We also examine the impact of the tumor microenvironment, including receptor expression and internalization, as well as VEGF secretion. We find that following anti-VEGF treatment, the concentration of free VEGF in the tumor can vary between 7 and 233 pM, with a dependence on both the density of VEGF receptors and co-receptors and the rate of neuropilin internalization on tumor cells. Finally, we predict that free VEGF in the tumor is reduced following anti-VEGF treatment when VEGF₁₂₁ comprises at least 25% of the VEGF secreted by tumor cells.

Conclusions

This study explores the optimal drug characteristics required for an anti-VEGF agent to have a therapeutic effect and the tumor-specific properties that influence the response to therapy. Our model provides a framework for investigating the use of VEGF-neutralizing drugs for personalized medicine treatment strategies.

Model-based global sensitivity analysis as applied to identification of anti-cancer drug targets and biomarkers of drug resistance in the ErbB2/3 network

High levels of variability in cancer-related cellular signalling networks and a lack of parameter identifiability in large-scale network models hamper translation of the results of modelling studies into the process of anti-cancer drug development. Recently global sensitivity analysis (GSA) has been recognised as a useful technique, capable of addressing the uncertainty of the model parameters and generating valid predictions on parametric sensitivities. Here we propose a novel implementation of model-based GSA specially designed to explore how multi-parametric network perturbations affect signal propagation through cancer-related networks. We use area-under-the-curve for time course of changes in phosphorylation of proteins as a characteristic for sensitivity analysis and rank network parameters with regard to their impact on the level of key cancer-related outputs, separating strong inhibitory from stimulatory effects. This allows interpretation of the results in terms which can incorporate the effects of potential anti-cancer drugs on targets and the associated biological markers of cancer. To illustrate the method we applied it to an ErbB signalling network model and explored the sensitivity profile of its key model readout, phosphorylated Akt, in the absence and presence of the ErbB2 inhibitor pertuzumab. The method successfully identified the parameters associated with elevation or suppression of Akt phosphorylation in the ErbB2/3 network. From analysis and comparison of the sensitivity profiles of pAkt in the absence and presence of targeted drugs we derived predictions of drug targets, cancer-related biomarkers and generated hypotheses for combinatorial therapy. Several key predictions have been confirmed in experiments using human ovarian carcinoma cell lines. We also compared GSA-derived predictions with the results of local sensitivity analysis and discuss the applicability of both methods. We propose that the developed GSA procedure can serve as a refining tool in combinatorial anti-cancer drug discovery.

Morphological changes in early melanoma development: influence of nutrients, growth inhibitors and cell-adhesion mechanisms

Current diagnostic methods for skin cancers are based on some morphological characteristics of the pigmented skin lesions, including the geometry of their contour. The aim of this article is to model the early growth of melanoma accounting for the biomechanical characteristics of the tumor micro-environment, and evaluating their influence on the tumor morphology and its evolution. The spatial distribution of tumor cells and diffusing molecules are explicitly described in a three-dimensional multiphase model, which incorporates general cell-to-cell mechanical interactions, a dependence of cell proliferation on contact inhibition, as well as a local diffusion of nutrients and inhibiting molecules. A two-dimensional model is derived in a lubrication limit accounting for the thin geometry of the epidermis. First, the dynamical and spatial properties of planar and circular tumor fronts are studied, with both numerical and analytical techniques. A WKB method is then developed in order to analyze the solution of the governing partial differential equations and to derive the threshold conditions for a contour instability of the growing tumor. A control parameter and a critical wavelength are identified, showing that high cell proliferation, high cell adhesion, large tumor radius and slow tumor growth correlate with the occurrence of a contour instability. Finally, comparing the theoretical results with a large amount of clinical data we show that our predictions describe accurately both the morphology of melanoma observed in vivo and its variations with the tumor growth rate. This study represents a fundamental step to understand more complex microstructural patterns observed during skin tumor growth. Its results have important implications for the improvement of the diagnostic methods for melanoma, possibly driving progress towards a personalized screening.

Systems biology of the metabolic network regulated by the Akt pathway

Cancer has been proposed as an example of systems biology disease or network disease. Accordingly, tumor cells differ from their normal counterparts more in terms of intracellular network dynamics than single markers. Here we shall focus on a recently recognized hallmark of cancer, the deregulation of cellular energetics. The constitutive activation of the phosphatidylinositol 3-kinase (PI3K)/Akt pathway has been confirmed as an essential step toward cell transformation. We will consider how the effects of Akt activation are connected with cell metabolism; more precisely, we will review existing metabolic models and discuss the current knowledge available to construct a kinetic model of the most relevant metabolic processes regulated by the PI3K/Akt pathway. The model will enable a systems biology approach to predict the metabolic targets that may inhibit cell growth under hyper activation of Akt.

Computational oncology

Oncology research has traditionally been conducted using techniques from the biological sciences. The new field of computational oncology has forged a new relationship between the physical sciences and oncology to further advance research. By applying physics and mathematics to oncologic problems, new insights will emerge into the pathogenesis and treatment of malignancies. One major area of investigation in computational oncology centers around the acquisition and analysis of data, using improved computing hardware and software. Large databases of cellular pathways are being analyzed to understand the interrelationship among complex biological processes. Computer-aided detection is being applied to the analysis of routine imaging data including mammography and chest imaging to improve the accuracy and detection rate for population screening. The second major area of investigation uses computers to construct sophisticated mathematical models of individual cancer cells as well as larger systems using partial differential equations. These models are further refined with clinically available information to more accurately reflect living systems. One of the major obstacles in the partnership between physical scientists and the oncology community is communications. Standard ways to convey information must be developed. Future progress in computational oncology will depend on close collaboration between clinicians and investigators to further the understanding of cancer using these new approaches.

Dynamic modelling of oestrogen signalling and cell fate in breast cancer cells

Cancers of the breast and other tissues arise from aberrant decision-making by cells regarding their survival or death, proliferation or quiescence, damage repair or bypass. These decisions are made by molecular signalling networks that process information from outside and from within the breast cancer cell and initiate responses that determine the cell's survival and reproduction. Because the molecular logic of these circuits is difficult to comprehend by intuitive reasoning alone, we present some preliminary mathematical models of the basic decision circuits in breast cancer cells that may aid our understanding of their susceptibility or resistance to endocrine therapy.

Hybrid models of tumor growth

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

Quantitative interpretation of a genetic model of carcinogenesis using computer simulations

The genetic model of tumorigenesis by Vogelstein et al. (V theory) and the molecular definition of cancer hallmarks by Hanahan and Weinberg (W theory) represent two of the most comprehensive and systemic understandings of cancer. Here, we develop a mathematical model that quantitatively interprets these seminal cancer theories, starting from a set of equations describing the short life cycle of an individual cell in uterine epithelium during tissue regeneration. The process of malignant transformation of an individual cell is followed and the tissue (or tumor) is described as a composite of individual cells in order to quantitatively account for intra-tumor heterogeneity. Our model describes normal tissue regeneration, malignant transformation, cancer incidence including dormant/transient tumors, and tumor evolution. Further, a novel mechanism for the initiation of metastasis resulting from substantial cell death is proposed. Finally, model simulations suggest two different mechanisms of metastatic inefficiency for aggressive and less aggressive cancer cells. Our work suggests that cellular de-differentiation is one major oncogenic pathway, a hypothesis based on a numerical description of a cell's differentiation status that can effectively and mathematically interpret some major concepts in V/W theories such as progressive transformation of normal cells, tumor evolution, and cancer hallmarks. Our model is a mathematical interpretation of cancer phenotypes that complements the well developed V/W theories based upon description of causal biological and molecular events. It is possible that further developments incorporating patient- and tissue-specific variables may build an even more comprehensive model to explain clinical observations and provide some novel insights for understanding cancer.

Exploiting clinical trial data drastically narrows the window of possible solutions to the problem of clinical adaptation of a multiscale cancer model

The development of computational models for simulating tumor growth and response to treatment has gained significant momentum during the last few decades. At the dawn of the era of personalized medicine, providing insight into complex mechanisms involved in cancer and contributing to patient-specific therapy optimization constitute particularly inspiring pursuits. The in silico oncology community is facing the great challenge of effectively translating simulation models into clinical practice, which presupposes a thorough sensitivity analysis, adaptation and validation process based on real clinical data. In this paper, the behavior of a clinically-oriented, multiscale model of solid tumor response to chemotherapy is investigated, using the paradigm of nephroblastoma response to preoperative chemotherapy in the context of the SIOP/GPOH clinical trial. A sorting of the model's parameters according to the magnitude of their effect on the output has unveiled the relative importance of the corresponding biological mechanisms; major impact on the result of therapy is credited to the oxygenation and nutrient availability status of the tumor and the balance between the symmetric and asymmetric modes of stem cell division. The effect of a number of parameter combinations on the extent of chemotherapy-induced tumor shrinkage and on the tumor's growth rate are discussed. A real clinical case of nephroblastoma has served as a proof of principle study case, demonstrating the basics of an ongoing clinical adaptation and validation process. By using clinical data in conjunction with plausible values of model parameters, an excellent fit of the model to the available medical data of the selected nephroblastoma case has been achieved, in terms of both volume reduction and histological constitution of the tumor. In this context, the exploitation of multiscale clinical data drastically narrows the window of possible solutions to the clinical adaptation problem.

Interoperability of time series cytometric data: a cross platform approach for modeling tumor heterogeneity

The cell cycle, with its highly conserved features, is a fundamental driver for the temporal control of cell proliferation-while abnormal control and modulation of the cell cycle are characteristic of tumor cells. The principal aim in cancer biology is to seek an understanding of the origin and nature of innate and acquired heterogeneity at the cellular level, driven principally by temporal and functional asynchrony. A major bottleneck when mathematically modeling these biological systems is the lack of interlinked structured experimental data. This often results in the in silico models failing to translate the specific hypothesis into parameterized terms that enable robust validation and hence would produce suitable prediction tools rather than just simulation tools. The focus has been on linking data originating from different cytometric platforms and cell-based event analysis to inform and constrain the input parameters of a compartmental cell cycle model, hence partly measuring and deconvolving cell cycle heterogeneity within a tumor population. Our work has addressed the concept that the interoperability of cytometric data, derived from different cytometry platforms, can complement as well as enhance cellular parameters space, thus providing a more broader and in-depth view of the cellular systems. The initial aim was to enable the cell cycle model to deliver an improved integrated simulation of the well-defined and constrained biological system. From a modeling perspective, such a cross platform approach has provided a paradigm shift from conventional cross-validation approaches, and from a bioinformatics perspective, novel computational methodology has been introduced for integrating and mapping continuous data with cross-sectional data. This establishes the foundation for developing predictive models and in silico tracking and prediction of tumor progression

Computational models of VEGF-associated angiogenic processes in cancer

Tumour angiogenesis allows a growing mass of cancer cells to overcome oxygen diffusion limitation and to increase cell survival. The growth of capillaries from pre-existing blood vessels is the result of numerous signalling cascades involving different molecules and of cellular events involving multiple cell and tissue types. Computational models offer insight into the mechanisms governing angiogenesis and provide quantitative information on parameters difficult to assess by experiments alone. In this article, we summarize results from computational models of tumour angiogenic processes with a focus on the molecular-detailed vascular endothelial growth factor-associated models that have been developed in our laboratory, spanning multiple scales from the molecular to whole body.

Physical oncology: a bench-to-bedside quantitative and predictive approach

Cancer models relating basic science to clinical care in oncology may fail to address the nuances of tumor behavior and therapy, as in the case, discussed herein, of the complex multiscale dynamics leading to the often-observed enhanced invasiveness, paradoxically induced by the very antiangiogenic therapy designed to destroy the tumor. Studies would benefit from approaches that quantitatively link the multiple physical and temporal scales from molecule to tissue in order to offer outcome predictions for individual patients. Physical oncology is an approach that applies fundamental principles from the physical and biological sciences to explain certain cancer behaviors as observable characteristics arising from the underlying physical and biochemical events. For example, the transport of oxygen molecules through tissue affects phenotypic characteristics such as cell proliferation, apoptosis, and adhesion, which in turn underlie the patient-scale tumor growth and invasiveness. Our review of physical oncology illustrates how tumor behavior and treatment response may be a quantifiable function of marginally stable molecular and/or cellular conditions modulated by inhomogeneity. By incorporating patient-specific genomic, proteomic, metabolomic, and cellular data into multiscale physical models, physical oncology could complement current clinical practice through enhanced understanding of cancer behavior, thus potentially improving patient survival.

Determination of somatic and cancer stem cell self-renewing symmetric division rate using sphere assays

Representing a renewable source for cell replacement, neural stem cells have received substantial attention in recent years. The neurosphere assay represents a method to detect the presence of neural stem cells, however owing to a deficiency of specific and definitive markers to identify them, their quantification and the rate they expand is still indefinite. Here we propose a mathematical interpretation of the neurosphere assay allowing actual measurement of neural stem cell symmetric division frequency. The algorithm of the modeling demonstrates a direct correlation between the overall cell fold expansion over time measured in the sphere assay and the rate stem cells expand via symmetric division. The model offers a methodology to evaluate specifically the effect of diseases and treatments on neural stem cell activity and function. Not only providing new insights in the evaluation of the kinetic features of neural stem cells, our modeling further contemplates cancer biology as cancer stem-like cells have been suggested to maintain tumor growth as somatic stem cells maintain tissue homeostasis. Indeed, tumor stem cell's resistance to therapy makes these cells a necessary target for effective treatment. The neurosphere assay mathematical model presented here allows the assessment of the rate malignant stem-like cells expand via symmetric division and the evaluation of the effects of therapeutics on the self-renewal and proliferative activity of this clinically relevant population that drive tumor growth and recurrence.

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

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

Identifying therapeutic targets in a combined EGFR-TGFβR signalling cascade using a multiscale agent-based cancer model

Applying a previously developed non-small cell lung cancer model, we assess 'cross-scale' the therapeutic efficacy of targeting a variety of molecular components of the epidermal growth factor receptor (EGFR) signalling pathway. Simulation of therapeutic inhibition and amplification allows for the ranking of the implemented downstream EGFR signalling molecules according to their therapeutic values or indices. Analysis identifies mitogen-activated protein kinase and extracellular signal-regulated kinase as top therapeutic targets for both inhibition and amplification-based treatment regimen but indicates that combined parameter perturbations do not necessarily improve the therapeutic effect of the separate parameter treatments as much as might be expected. Potential future strategies using this in silico model to tailor molecular treatment regimen are discussed.

Increase of plasma VEGF after intravenous administration of bevacizumab is predicted by a pharmacokinetic model

Vascular endothelial growth factor (VEGF) is one of the most potent cytokines targeted in antiangiogenic therapies. Bevacizumab, a recombinant humanized monoclonal antibody to VEGF, is being used clinically in combination with chemotherapy for colorectal, non-small cell lung and breast cancers, and as a single agent for glioblastoma and is being tested for other types of cancer in numerous clinical trials. It has been reported that the intravenous injection of bevacizumab leads to an increase of plasma VEGF concentration in cancer patients. The mechanism responsible for this counterintuitive increase has not been elucidated, although several hypotheses have been proposed. We use a multiscale systems biology approach to address this problem. We have constructed a whole-body pharmacokinetic model comprising three compartments: blood, normal tissue, and tumor tissue. Molecular interactions among VEGF-A family members, their major receptors, the extracellular matrix, and an anti-VEGF ligand are considered for each compartment. Diffusible molecules extravasate, intravasate, are removed from the healthy tissue through the lymphatics, and are cleared from the blood.

Theory of hybrid dynamical systems and its applications to biological and medical systems

In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of continuous and discrete dynamics. It is now well known that there exist many hybrid dynamical systems with discontinuities such as impact, switching, friction and sliding. The first aim of this Issue is to discuss recent developments in understanding nonlinear dynamics of hybrid dynamical systems in the two main theoretical fields of dynamical systems theory and control systems theory. A combined study of the hybrid systems dynamics in the two theoretical fields might contribute to a more comprehensive understanding of hybrid dynamical systems. In addition, mathematical modelling by hybrid dynamical systems is particularly important for understanding the nonlinear dynamics of biological and medical systems as they have many discontinuities such as threshold-triggered firing in neurons, on-off switching of gene expression by a transcription factor, division in cells and certain types of chronotherapy for prostate cancer. Hence, the second aim is to discuss recent applications of hybrid dynamical systems in biology and medicine. Thus, this Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine. As the introduction to the topics in this Theme Issue, we provide a brief history of nonlinear dynamics and mathematical modelling, different mathematical models of hybrid dynamical systems, the relationship between dynamical systems theory and control systems theory, examples of complex behaviour in a simple neuron model and its variants, applications of hybrid dynamical systems in biology and medicine as a road map of articles in this Theme Issue and future directions of hybrid systems modelling.

A multiscale systems perspective on cancer, immunotherapy, and Interleukin-12

Monoclonal antibodies represent some of the most promising molecular targeted immunotherapies. However, understanding mechanisms by which tumors evade elimination by the immune system of the host presents a significant challenge for developing effective cancer immunotherapies. The interaction of cancer cells with the host is a complex process that is distributed across a variety of time and length scales. The time scales range from the dynamics of protein refolding (i.e., microseconds) to the dynamics of disease progression (i.e., years). The length scales span the farthest reaches of the human body (i.e., meters) down to the range of molecular interactions (i.e., nanometers). Limited ranges of time and length scales are used experimentally to observe and quantify changes in physiology due to cancer. Translating knowledge obtained from the limited scales observed experimentally to predict patient response is an essential prerequisite for the rational design of cancer immunotherapies that improve clinical outcomes. In studying multiscale systems, engineers use systems analysis and design to identify important components in a complex system and to test conceptual understanding of the integrated system behavior using simulation. The objective of this review is to summarize interactions between the tumor and cell-mediated immunity from a multiscale perspective. Interleukin-12 and its role in coordinating antibody-dependent cell-mediated cytotoxicity is used illustrate the different time and length scale that underpin cancer immunoediting. An underlying theme in this review is the potential role that simulation can play in translating knowledge across scales.

Information technology solutions for integration of biomolecular and clinical data in the identification of new cancer biomarkers and targets for therapy

The quest for new cancer biomarkers and targets for therapy requires not only the aggregation and analysis of heterogeneous biomolecular data but also integration of clinical data. In this review we highlight information technology solutions for the integration of biomolecular and clinical data and focus on a solution at the departmental level, i.e., decentralized and medium-scale solution for groups of labs working on a specific topic. Both, hardware and software requirements are described as well as bioinformatics methods and tools for the data analysis. The highlighted IT solutions include storage architecture, high-performance computing, and application servers. Additionally, following computational approaches for data integration are reviewed: data aggregation, integrative data analysis including methodological aspects as well as examples, biomolecular pathways and network reconstruction, and mathematical modelling. Finally, a case study in cancer immunology including the used computational methods is shown, demonstrating how IT solutions for integrating biomolecular and clinical data can help to identify new cancer biomarkers for improving diagnosis and predicting clinical outcome.

Modeling tumor growth and treatment response based on quantitative imaging data

We review current approaches to predicting tumor growth and treatment response that combine non-invasive imaging data with mathematical models of cancer progression, and propose some new directions for integrating quantitative imaging measurements with such numerical analyses. Historically, tumor modeling has been described by parameters that are measurable by invasive methods only or in isolated in vitro or ex vivo systems. This limits the practical usefulness of such models because it is not possible to test their predictions experimentally. Recent advances in three-dimensional magnetic resonance imaging, single photon emission computed tomography, and positron emission tomography techniques provide new opportunities to acquire measurements of relevant molecular and cellular features of tumors non-invasively and with high spatial resolution. Such data can be incorporated into mathematical models of tumors. We highlight some recent examples of this approach and identify several simple examples that allow for conventional mathematical models of tumor growth to be recast in terms of parameters that can be measured by imaging, thus raising the possibility of designing and constraining models that can be tested in clinical practice. It is our hope that this Perspective will stimulate further work in this evolving and exciting field.

Predictions of tumour morphological stability and evaluation against experimental observations

The hallmark of malignant tumours is their spread into neighbouring tissue and metastasis to distant organs, which can lead to life threatening consequences. One of the defining characteristics of aggressive tumours is an unstable morphology, including the formation of invasive fingers and protrusions observed both in vitro and in vivo. In spite of extensive biological, clinical and modelling study and research at physical scales ranging from the molecular to the tissue, the driving dynamics of tumour invasiveness are not completely understood, partly because it is challenging to observe and study cancer as a multi-scale system. Mathematical modelling has been applied to provide further insights into these complex invasive and metastatic behaviours. Modelling a solid tumour as an incompressible fluid, we consider three possible constitutive relations to describe tumour growth, namely Darcy's law, Stokes' law and the combined Darcy-Stokes law. We study the tumour morphological stability described by each model and evaluate the consistency between theoretical model predictions and experimental data from in vitro three-dimensional multicellular tumour spheroids. The analysis reveals that the Stokes model is the most consistent with the experimental observations, and that it predicts our experimental tumour growth is marginally stable. We further show that it is feasible to extract parameter values from a limited set of data and create a self-consistent modelling framework that can be extended to the multi-scale study of cancer.

Streaming instability in growing cell populations

Flows of cells growing as a quasimonolayer in a confined space can exhibit streaming, with narrow streams of fast-moving cells flowing around clusters of slowly moving cells. We observed and analyzed this phenomenon experimentally for E. coli bacteria proliferating in a microfluidic cell trap using time-lapse microscopy. We also performed continuum modeling and discrete-element simulations to elucidate the mechanism behind the streaming instability. Our analysis demonstrates that streaming can be explained by the interplay between the slow adaptation of a cell to its local microenvironment and its mobility due to changes of cell-substrate contact forces.

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.

Selection in spatial stochastic models of cancer: migration as a key modulator of fitness

Background

We study the selection dynamics in a heterogeneous spatial colony of cells. We use two spatial generalizations of the Moran process, which include cell divisions, death and migration. In the first model, migration is included explicitly as movement to a proximal location. In the second, migration is implicit, through the varied ability of cell types to place their offspring a distance away, in response to another cell's death.

Results

In both models, we find that migration has a direct positive impact on the ability of a single mutant cell to invade a pre-existing colony. Thus, a decrease in the growth potential can be compensated by an increase in cell migration. We further find that the neutral ridges (the set of all types with the invasion probability equal to that of the host cells) remain invariant under the increase of system size (for large system sizes), thus making the invasion probability a universal characteristic of the cells selection status. We find that repeated instances of large scale cell-death, such as might arise during therapeutic intervention or host response, strongly select for the migratory phenotype.

Conclusions

These models can help explain the many examples in the biological literature, where genes involved in cell's migratory and invasive machinery are also associated with increased cellular fitness, even though there is no known direct effect of these genes on the cellular reproduction. The models can also help to explain how chemotherapy may provide a selection mechanism for highly invasive phenotypes.

Reviewers

This article was reviewed by Marek Kimmel and Glenn Webb.

Three-dimensional multispecies nonlinear tumor growth-II: Tumor invasion and angiogenesis

We extend the diffuse interface model developed in Wise et al. (2008) to study nonlinear tumor growth in 3-D. Extensions include the tracking of multiple viable cell species populations through a continuum diffuse-interface method, onset and aging of discrete tumor vessels through angiogenesis, and incorporation of individual cell movement using a hybrid continuum-discrete approach. We investigate disease progression as a function of cellular-scale parameters such as proliferation and oxygen/nutrient uptake rates. We find that heterogeneity in the physiologically complex tumor microenvironment, caused by non-uniform distribution of oxygen, cell nutrients, and metabolites, as well as phenotypic changes affecting cellular-scale parameters, can be quantitatively linked to the tumor macro-scale as a mechanism that promotes morphological instability. This instability leads to invasion through tumor infiltration of surrounding healthy tissue. Models that employ a biologically founded, multiscale approach, as illustrated in this work, could help to quantitatively link the critical effect of heterogeneity in the tumor microenvironment with clinically observed tumor growth and invasion. Using patient tumor-specific parameter values, this may provide a predictive tool to characterize the complex in vivo tumor physiological characteristics and clinical response, and thus lead to improved treatment modalities and prognosis.

An off-lattice hybrid discrete-continuum model of tumor growth and invasion

We have developed an off-lattice hybrid discrete-continuum (OLHDC) model of tumor growth and invasion. The continuum part of the OLHDC model describes microenvironmental components such as matrix-degrading enzymes, nutrients or oxygen, and extracellular matrix (ECM) concentrations, whereas the discrete portion represents individual cell behavior such as cell cycle, cell-cell, and cell-ECM interactions and cell motility by the often-used persistent random walk, which can be depicted by the Langevin equation. Using this framework of the OLHDC model, we develop a phenomenologically realistic and bio/physically relevant model that encompasses the experimentally observed superdiffusive behavior (at short times) of mammalian cells. When systemic simulations based on the OLHDC model are performed, tumor growth and its morphology are found to be strongly affected by cell-cell adhesion and haptotaxis. There is a combination of the strength of cell-cell adhesion and haptotaxis in which fingerlike shapes, characteristic of invasive tumor, are observed.

Data integration and exploration for the identification of molecular mechanisms in tumor-immune cells interaction

Cancer progression is a complex process involving host-tumor interactions by multiple molecular and cellular factors of the tumor microenvironment. Tumor cells that challenge immune activity may be vulnerable to immune destruction. To address this question we have directed major efforts towards data integration and developed and installed a database for cancer immunology with more than 1700 patients and associated clinical data and biomolecular data. Mining of the database revealed novel insights into the molecular mechanisms of tumor-immune cell interaction. In this paper we present the computational tools used to analyze integrated clinical and biomolecular data. Specifically, we describe a database for heterogeneous data types, the interfacing bioinformatics and statistical tools including clustering methods, survival analysis, as well as visualization methods. Additionally, we discuss generic issues relevant to the integration of clinical and biomolecular data, as well as recent developments in integrative data analyses including biomolecular network reconstruction and mathematical modeling.

Perspectives on tissue interactions in development and disease

From the morphogenetic movements of the three germ layers during development to the reactive stromal microenvironment in cancer, tissue interactions are vital to maintaining healthy organ morphologic architecture and function. The stromal compartment is thought to be complicit in tumor progression and, as such, represents an opportune target for disease therapies. However, recent developments in our understanding of the diversity of the stromal compartment and the lack of appropriate models to study its relevance in human disease have limited our further understanding of the role of tissue interactions in tumor progression. The failure any model to fully recapitulate the complexities of systemic biology continue to create a higher imperative for incorporating various perspectives into a broader understanding for the ultimate goal of designing interventional therapies. Understanding this potential, this review examines the biological models used to study stromal-epithelial interactions and includes an attempt to incorporate behavioral terminology to define and mathematically model ecological relationships in stromal-epithelial interactions. In addition, the current attempt to incorporate these diverse ecological perspectives into in silico mathematical models through cross-disciplinary coordination is reviewed, which will provide a fresh perspective on defining cell group behavior and tissue ecology in disease and hopefully lead to the generation of new hypotheses to be empirically validated.

Cancer stem cell tumor model reveals invasive morphology and increased phenotypical heterogeneity

The recently developed concept of cancer stem cells (CSC) sheds new light on various aspects of tumor growth and progression. Here, we present a mathematical model of malignancies to investigate how a hierarchical organized cancer cell population affects the fundamental properties of solid malignancies. We establish that tumors modeled in a CSC context more faithfully resemble human malignancies and show invasive behavior, whereas tumors without a CSC hierarchy do not. These findings are corroborated by in vitro studies. In addition, we provide evidence that the CSC model is accompanied by highly altered evolutionary dynamics compared with the ones predicted to exist in a stochastic, nonhierarchical tumor model. Our main findings indicate that the CSC model allows for significantly higher tumor heterogeneity, which may affect therapy resistance. Moreover, we show that therapy which fails to target the CSC population is not only unsuccessful in curing the patient, but also promotes malignant features in the recurring tumor. These include rapid expansion, increased invasion, and enhanced heterogeneity.

Cancer systems biology: a network modeling perspective

Cancer is now appreciated as not only a highly heterogenous pathology with respect to cell type and tissue origin but also as a disease involving dysregulation of multiple pathways governing fundamental cell processes such as death, proliferation, differentiation and migration. Thus, the activities of molecular networks that execute metabolic or cytoskeletal processes, or regulate these by signal transduction, are altered in a complex manner by diverse genetic mutations in concert with the environmental context. A major challenge therefore is how to develop actionable understanding of this multivariate dysregulation, with respect both to how it arises from diverse genetic mutations and to how it may be ameliorated by prospective treatments. While high-throughput experimental platform technologies ranging from genomic sequencing to transcriptomic, proteomic and metabolomic profiling are now commonly used for molecular-level characterization of tumor cells and surrounding tissues, the resulting data sets defy straightforward intuitive interpretation with respect to potential therapeutic targets or the effects of perturbation. In this review article, we will discuss how significant advances can be obtained by applying computational modeling approaches to elucidate the pathways most critically involved in tumor formation and progression, impact of particular mutations on pathway operation, consequences of altered cell behavior in tissue environments and effects of molecular therapeutics.

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

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

An empirical Bayesian approach for model-based inference of cellular signaling networks

Background

A common challenge in systems biology is to infer mechanistic descriptions of biological process given limited observations of a biological system. Mathematical models are frequently used to represent a belief about the causal relationships among proteins within a signaling network. Bayesian methods provide an attractive framework for inferring the validity of those beliefs in the context of the available data. However, efficient sampling of high-dimensional parameter space and appropriate convergence criteria provide barriers for implementing an empirical Bayesian approach. The objective of this study was to apply an Adaptive Markov chain Monte Carlo technique to a typical study of cellular signaling pathways.

Results

As an illustrative example, a kinetic model for the early signaling events associated with the epidermal growth factor (EGF) signaling network was calibrated against dynamic measurements observed in primary rat hepatocytes. A convergence criterion, based upon the Gelman-Rubin potential scale reduction factor, was applied to the model predictions. The posterior distributions of the parameters exhibited complicated structure, including significant covariance between specific parameters and a broad range of variance among the parameters. The model predictions, in contrast, were narrowly distributed and were used to identify areas of agreement among a collection of experimental studies.

Conclusion

In summary, an empirical Bayesian approach was developed for inferring the confidence that one can place in a particular model that describes signal transduction mechanisms and for inferring inconsistencies in experimental measurements.

Cells, cancer, and rare events: homeostatic metastability in stochastic nonlinear dynamical models of skin cell proliferation

A recently proposed model for skin cell proliferation [E. Clayton, Nature (London) 446, 185 (2007)] is extended to incorporate mitotic autoregulation, and hence homeostasis as a fixed point of the dynamics. Unlimited cell proliferation in such a model can be viewed as a model for carcinogenesis. One way in which this can arise is homeostatic metastability, in which the cell populations escape from the homeostatic basin of attraction by a large but rare stochastic fluctuation. Such an event can be viewed as the final step in a multistage model of carcinogenesis. Homeostatic metastability offers a possible explanation for the peculiar epidemiology of lung cancer in ex-smokers.

Fractal analysis and ionic dependence of endocytotic membrane activity of human breast cancer cells

The endocytic membrane activities of two human breast cancer cell lines (MDA-MB-231 and MCF-7) of strong and weak metastatic potential, respectively, were studied in a comparative approach. Uptake of horseradish peroxidase was used to follow endocytosis. Dependence on ionic conditions and voltage-gated sodium channel (VGSC) activity were characterized. Fractal methods were used to analyze quantitative differences in vesicular patterning. Digital quantification showed that MDA-MB-231 cells took up more tracer (i.e., were more endocytic) than MCF-7 cells. For the former, uptake was totally dependent on extracellular Na(+) and partially dependent on extracellular and intracellular Ca(2+) and protein kinase activity. Analyzing the generalized fractal dimension (D(q )) and its Legendre transform f(alpha) revealed that under control conditions, all multifractal parameters determined had values greater for MDA-MB-231 compared with MCF-7 cells, consistent with endocytic/vesicular activity being more developed in the strongly metastatic cells. All fractal parameters studied were sensitive to the VGSC blocker tetrodotoxin (TTX). Some of the parameters had a "simple" dependence on VGSC activity, if present, whereby pretreatment with TTX reduced the values for the MDA-MB-231 cells and eliminated the differences between the two cell lines. For other parameters, however, there was a "complex" dependence on VGSC activity. The possible physical/physiological meaning of the mathematical parameters studied and the nature of involvement of VGSC activity in control of endocytosis/secretion are discussed.