Integrative mathematical oncology

A versatile mathematical work-flow to explore how Cancer Stem Cell fate influences tumor progression

Background

Nowadays multidisciplinary approaches combining mathematical models with experimental assays are becoming relevant for the study of biological systems. Indeed, in cancer research multidisciplinary approaches are successfully used to understand the crucial aspects implicated in tumor growth. In particular, the Cancer Stem Cell (CSC) biology represents an area particularly suited to be studied through multidisciplinary approaches, and modeling has significantly contributed to pinpoint the crucial aspects implicated in this theory.

More generally, to acquire new insights on a biological system it is necessary to have an accurate description of the phenomenon, such that making accurate predictions on its future behaviors becomes more likely. In this context, the identification of the parameters influencing model dynamics can be advantageous to increase model accuracy and to provide hints in designing wet experiments. Different techniques, ranging from statistical methods to analytical studies, have been developed. Their applications depend on case-specific aspects, such as the availability and quality of experimental data, and the dimension of the parameter space.

Results

The study of a new model on the CSC-based tumor progression has been the motivation to design a new work-flow that helps to characterize possible system dynamics and to identify those parameters influencing such behaviors. In detail, we extended our recent model on CSC-dynamics creating a new system capable of describing tumor growth during the different stages of cancer progression. Indeed, tumor cells appear to progress through lineage stages like those of normal tissues, being their division auto-regulated by internal feedback mechanisms. These new features have introduced some non-linearities in the model, making it more difficult to be studied by solely analytical techniques. Our new work-flow, based on statistical methods, was used to identify the parameters which influence the tumor growth. The effectiveness of the presented work-flow was firstly verified on two well known models and then applied to investigate our extended CSC model.

Conclusions

We propose a new work-flow to study in a practical and informative way complex systems, allowing an easy identification, interpretation, and visualization of the key model parameters. Our methodology is useful to investigate possible model behaviors and to establish factors driving model dynamics.

Analyzing our new CSC model guided by the proposed work-flow, we found that the deregulation of CSC asymmetric proliferation contributes to cancer initiation, in accordance with several experimental evidences. Specifically, model results indicated that the probability of CSC symmetric proliferation is responsible of a switching-like behavior which discriminates between tumorigenesis and unsustainable tumor growth.

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

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

Can a minimal replicating construct be identified as the embodiment of cancer?

Genomic instability is a hallmark of cancer. Cancer cells that exhibit abnormal chromosomes are characteristic of most advanced tumours, despite the potential threat represented by accumulated genetic damage. Carcinogenesis involves a loss of key components of the genetic and signalling molecular networks; hence some authors have suggested that this is part of a trend of cancer cells to behave as simple, minimal replicators. In this study, we explore this conjecture and suggest that, in the case of cancer, genomic instability has an upper limit that is associated with a minimal cancer cell network. Such a network would include (for a given microenvironment) the basic molecular components that allow cells to replicate and respond to selective pressures. However, it would also exhibit internal fragilities that could be exploited by appropriate therapies targeting the DNA repair machinery. The implications of this hypothesis are discussed.

Systems biology network-based discovery of a small molecule activator BL-AD008 targeting AMPK/ZIPK and inducing apoptosis in cervical cancer

The aim of this study was to discover a small molecule activator BL-AD008 targeting AMPK/ZIPK and inducing apoptosis in cervical cancer. In this study, we systematically constructed the global protein-protein interaction (PPI) network and predicted apoptosis-related protein connections by the Naïve Bayesian model. Then, we identified some classical apoptotic PPIs and other previously unrecognized PPIs between apoptotic kinases, such as AMPK and ZIPK. Subsequently, we screened a series of candidate compounds targeting AMPK/ZIPK, synthesized some compounds and eventually discovered a novel dual-target activator (BL-AD008). Moreover, we found BL-AD008 bear remarkable anti-proliferative activities toward cervical cancer cells and could induce apoptosis by death-receptor and mitochondrial pathways. Additionally, we found that BL-AD008-induced apoptosis was affected by the combination of AMPK and ZIPK. Then, we found that BL-AD008 bear its anti-tumor activities and induced apoptosis by targeting AMPK/ZIPK in vivo. In conclusion, these results demonstrate the ability of systems biology network to identify some key apoptotic kinase targets AMPK and ZIPK; thus providing a dual-target small molecule activator (BL-AD008) as a potential new apoptosis-modulating drug in future cervical cancer therapy.

Bridging scales in cancer progression: mapping genotype to phenotype using neural networks

In this review we summarise our recent efforts in trying to understand the role of heterogeneity in cancer progression by using neural networks to characterise different aspects of the mapping from a cancer cells genotype and environment to its phenotype. Our central premise is that cancer is an evolving system subject to mutation and selection, and the primary conduit for these processes to occur is the cancer cell whose behaviour is regulated on multiple biological scales. The selection pressure is mainly driven by the microenvironment that the tumour is growing in and this acts directly upon the cell phenotype. In turn, the phenotype is driven by the intracellular pathways that are regulated by the genotype. Integrating all of these processes is a massive undertaking and requires bridging many biological scales (i.e. genotype, pathway, phenotype and environment) that we will only scratch the surface of in this review. We will focus on models that use neural networks as a means of connecting these different biological scales, since they allow us to easily create heterogeneity for selection to act upon and importantly this heterogeneity can be implemented at different biological scales. More specifically, we consider three different neural networks that bridge different aspects of these scales and the dialogue with the micro-environment, (i) the impact of the micro-environment on evolutionary dynamics, (ii) the mapping from genotype to phenotype under drug-induced perturbations and (iii) pathway activity in both normal and cancer cells under different micro-environmental conditions.

Cancer evolution: mathematical models and computational inference

Cancer is a somatic evolutionary process characterized by the accumulation of mutations, which contribute to tumor growth, clinical progression, immune escape, and drug resistance development. Evolutionary theory can be used to analyze the dynamics of tumor cell populations and to make inference about the evolutionary history of a tumor from molecular data. We review recent approaches to modeling the evolution of cancer, including population dynamics models of tumor initiation and progression, phylogenetic methods to model the evolutionary relationship between tumor subclones, and probabilistic graphical models to describe dependencies among mutations. Evolutionary modeling helps to understand how tumors arise and will also play an increasingly important prognostic role in predicting disease progression and the outcome of medical interventions, such as targeted therapy.

Modeling intrinsic heterogeneity and growth of cancer cells

Intratumoral heterogeneity has been found to be a major cause of drug resistance. Cell-to-cell variation increases as a result of cancer-related alterations, which are acquired by stochastic events and further induced by environmental signals. However, most cellular mechanisms include natural fluctuations that are closely regulated, and thus lead to asynchronization of the cells, which causes intrinsic heterogeneity in a given population. Here, we derive two novel mathematical models, a stochastic agent-based model and an integro-differential equation model, each of which describes the growth of cancer cells as a dynamic transition between proliferative and quiescent states. These models are designed to predict variations in growth as a function of the intrinsic heterogeneity emerging from the durations of the cell-cycle and apoptosis, and also include cellular density dependencies. By examining the role all parameters play in the evolution of intrinsic tumor heterogeneity, and the sensitivity of the population growth to parameter values, we show that the cell-cycle length has the most significant effect on the growth dynamics. In addition, we demonstrate that the agent-based model can be approximated well by the more computationally efficient integro-differential equations when the number of cells is large. This essential step in cancer growth modeling will allow us to revisit the mechanisms of multidrug resistance by examining spatiotemporal differences of cell growth while administering a drug among the different sub-populations in a single tumor, as well as the evolution of those mechanisms as a function of the resistance level.

Exploring the competition between proliferative and invasive cancer phenotypes in a continuous spatial model

Tumor is characterized by extensive heterogeneity with respect to its microenvironment and its genetic composition. We extend a previously developed monoclonal continuous spatial model of tumor growth to account for polyclonal cell populations and investigate the interplay between a more proliferative and a more invasive phenotype under different conditions. The model simulations demonstrate a transition from the dominance of the proliferative to the dominance of the invasive phenotype resembling malignant tumor progression and show a time period where both subpopulations are abundant. As the dominant phenotype switches from proliferative to invasive, the geometry of tumor changes from a compact and almost spherical shape to a more diffusive and fingered morphology with the proliferative phenotype to be restricted in the tumor bulk and the invasive to dominate at tumor edges. Different micro-environmental conditions and different phenotypic properties can promote or inhibit invasion demonstrating their mutual importance. The model provides a computational framework to investigate tumor heterogeneity and the constant interplay between the environment and the specific characteristics of phenotypes that should be taken into account for the prediction of tumor evolution, morphology and effective treatment.

Mathematical modeling for novel cancer drug discovery and development

INTRODUCTION

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

AREAS COVERED

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

EXPERT OPINION

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

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

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

An integrated computational model of the bone microenvironment in bone-metastatic prostate cancer

Bone metastasis will impact most men with advanced prostate cancer. The vicious cycle of bone degradation and formation driven by metastatic prostate cells in bone yields factors that drive cancer growth. Mechanistic insights into this vicious cycle have suggested new therapeutic opportunities, but complex temporal and cellular interactions in the bone microenvironment make drug development challenging. We have integrated biologic and computational approaches to generate a hybrid cellular automata model of normal bone matrix homeostasis and the prostate cancer-bone microenvironment. The model accurately reproduces the basic multicellular unit bone coupling process, such that introduction of a single prostate cancer cell yields a vicious cycle similar in cellular composition and pathophysiology to models of prostate-to-bone metastasis. Notably, the model revealed distinct phases of osteolytic and osteogenic activity, a critical role for mesenchymal stromal cells in osteogenesis, and temporal changes in cellular composition. To evaluate the robustness of the model, we assessed the effect of established bisphosphonate and anti-RANKL therapies on bone metastases. At approximately 100% efficacy, bisphosphonates inhibited cancer progression while, in contrast with clinical observations in humans, anti-RANKL therapy fully eradicated metastases. Reducing anti-RANKL yielded clinically similar results, suggesting that better targeting or dosing could improve patient survival. Our work establishes a computational model that can be tailored for rapid assessment of experimental therapies and delivery of precision medicine to patients with prostate cancer with bone metastases.

A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane

In this paper, we extend the 3D multispecies diffuse-interface model of the tumor growth, which was derived in Wise et al. (Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524-543), and incorporate the effect of a stiff membrane to model tumor growth in a confined microenvironment. We then develop accurate and efficient numerical methods to solve the model. When the membrane is endowed with a surface energy, the model is variational, and the numerical scheme, which involves adaptive mesh refinement and a nonlinear multigrid finite difference method, is demonstrably shown to be energy stable. Namely, in the absence of cell proliferation and death, the discrete energy is a nonincreasing function of time for any time and space steps. When a simplified model of membrane elastic energy is used, the resulting model is derived analogously to the surface energy case. However, the elastic energy model is actually nonvariational because certain coupling terms are neglected. Nevertheless, a very stable numerical scheme is developed following the strategy used in the surface energy case. 2D and 3D simulations are performed that demonstrate the accuracy of the algorithm and illustrate the shape instabilities and nonlinear effects of membrane elastic forces that may resist or enhance growth of the tumor. Compared with the standard Crank-Nicholson method, the time step can be up to 25 times larger using the new approach.

Exploiting ecological principles to better understand cancer progression and treatment

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

Wnt-responsive cancer stem cells are located close to distorted blood vessels and not in hypoxic regions in a p53-null mouse model of human breast cancer

Cancer stem cells (CSCs, or tumor-initiating cells) may be responsible for tumor formation in many types of cancer, including breast cancer. Using high-resolution imaging techniques, we analyzed the relationship between a Wnt-responsive, CSC-enriched population and the tumor vasculature using p53-null mouse mammary tumors transduced with a lentiviral Wnt signaling reporter. Consistent with their localization in the normal mammary gland, Wnt-responsive cells in tumors were enriched in the basal/myoepithelial population and generally located in close proximity to blood vessels. The Wnt-responsive CSCs did not colocalize with the hypoxia-inducible factor 1α-positive cells in these p53-null basal-like tumors. Average vessel diameter and vessel tortuosity were increased in p53-null mouse tumors, as well as in a human tumor xenograft as compared with the normal mammary gland. The combined strategy of monitoring the fluorescently labeled CSCs and vasculature using high-resolution imaging techniques provides a unique opportunity to study the CSC and its surrounding vasculature.

In vitro models of tumor vessels and matrix: engineering approaches to investigate transport limitations and drug delivery in cancer

Tumor-stroma interactions have emerged as critical determinants of drug efficacy. However, the underlying biological and physicochemical mechanisms by which the microenvironment regulates therapeutic response remain unclear, due in part to a lack of physiologically relevant in vitro platforms to accurately interrogate tissue-level phenomena. Tissue-engineered tumor models are beginning to address this shortcoming. By allowing selective incorporation of microenvironmental complexity, these platforms afford unique access to tumor-associated signaling and transport dynamics. This review will focus on engineering approaches to study drug delivery as a function of tumor-associated changes of the vasculature and extracellular matrix (ECM). First, we review current biological understanding of these components and discuss their impact on transport processes. Then, we evaluate existing microfluidic, tissue engineering, and materials science strategies to recapitulate vascular and ECM characteristics of tumors, and finish by outlining challenges and future directions of the field that may ultimately improve anti-cancer therapies.

Estimating dose painting effects in radiotherapy: a mathematical model

Tumor heterogeneity is widely considered to be a determinant factor in tumor progression and in particular in its recurrence after therapy. Unfortunately, current medical techniques are unable to deduce clinically relevant information about tumor heterogeneity by means of non-invasive methods. As a consequence, when radiotherapy is used as a treatment of choice, radiation dosimetries are prescribed under the assumption that the malignancy targeted is of a homogeneous nature. In this work we discuss the effects of different radiation dose distributions on heterogeneous tumors by means of an individual cell-based model. To that end, a case is considered where two tumor cell phenotypes are present, which we assume to strongly differ in their respective cell cycle duration and radiosensitivity properties. We show herein that, as a result of such differences, the spatial distribution of the corresponding phenotypes, whence the resulting tumor heterogeneity can be predicted as growth proceeds. In particular, we show that if we start from a situation where a majority of ordinary cancer cells (CCs) and a minority of cancer stem cells (CSCs) are randomly distributed, and we assume that the length of CSC cycle is significantly longer than that of CCs, then CSCs become concentrated at an inner region as tumor grows. As a consequence we obtain that if CSCs are assumed to be more resistant to radiation than CCs, heterogeneous dosimetries can be selected to enhance tumor control by boosting radiation in the region occupied by the more radioresistant tumor cell phenotype. It is also shown that, when compared with homogeneous dose distributions as those being currently delivered in clinical practice, such heterogeneous radiation dosimetries fare always better than their homogeneous counterparts. Finally, limitations to our assumptions and their resulting clinical implications will be discussed.

The effects of cell compressibility, motility and contact inhibition on the growth of tumor cell clusters using the Cellular Potts Model

There are numerous biological examples where genes associated with migratory ability of cells also confer the cells with an increased fitness even though these genes may not have any known effect on the cell mitosis rates. Here, we provide insight into these observations by analyzing the effects of cell migration, compression, and contact inhibition on the growth of tumor cell clusters using the Cellular Potts Model (CPM) in a monolayer geometry. This is a follow-up of a previous study (Thalhauser et al. 2010) in which a Moran-type model was used to study the interaction of cell proliferation, migratory potential and death on the emergence of invasive phenotypes. Here, we extend the study to include the effects of cell size and shape. In particular, we investigate the interplay between cell motility and compressibility within the CPM and find that the CPM predicts that increased cell motility leads to smaller cells. This is an artifact in the CPM. An analysis of the CPM reveals an explicit inverse-relationship between the cell stiffness and motility parameters. We use this relationship to compensate for motility-induced changes in cell size in the CPM so that in the corrected CPM, cell size is independent of the cell motility. We find that subject to comparable levels of compression, clusters of motile cells grow faster than clusters of less motile cells, in qualitative agreement with biological observations and our previous study. Increasing compression tends to reduce growth rates. Contact inhibition penalizes clumped cells by halting their growth and gives motile cells an even greater advantage. Finally, our model predicts cell size distributions that are consistent with those observed in clusters of neuroblastoma cells cultured in low and high density conditions.

Tumour-stromal interactions generate emergent persistence in collective cancer cell migration

Cancer cell collective migration is a complex behaviour leading to the invasion of cancer cells into surrounding tissue, often with the aid of stromal cells in the microenvironment, such as macrophages or fibroblasts. Although tumour-tumour and tumour-stromal intercellular signalling have been shown to contribute to cancer cell migration, we lack a fundamental theoretical understanding of how aggressive invasion emerges from the synergy between these mechanisms. We use a computational self-propelled particle model to simulate intercellular interactions between co-migrating tumour and stromal cells and study the emergence of collective movement. We find that tumour-stromal interaction increases the cohesion and persistence of migrating mixed tumour-stromal cell clusters in a noisy and unbounded environment, leading to increased cell cluster size and distance migrated by cancer cells. Although environmental constraints, such as vasculature or extracellular matrix, influence cancer migration in vivo, our model shows that cell-cell interactions are sufficient to generate cohesive and persistent movement. From our results, we conclude that inhibition of tumour-stromal intercellular signalling may present a viable therapeutic target for disrupting collective cancer cell migration.

Spatial invasion dynamics on random and unstructured meshes: implications for heterogeneous tumor populations

In this work we discuss a spatial evolutionary model for a heterogeneous cancer cell population. We consider the gain-of-function mutations that not only change the fitness potential of the mutant phenotypes against normal background cells but may also increase the relative motility of the mutant cells. The spatial modeling is implemented as a stochastic evolutionary system on a structured grid (a lattice, with random neighborhoods, which is not necessarily bi-directional) or on a two-dimensional unstructured mesh, i.e. a bi-directional graph with random numbers of neighbors. We present a computational approach to investigate the fixation probability of mutants in these spatial models. Additionally, we examine the effect of the migration potential on the spatial dynamics of mutants on unstructured meshes. Our results suggest that the probability of fixation is negatively correlated with the width of the distribution of the neighborhood size. Also, the fixation probability increases given a migration potential for mutants. We find that the fixation probability (of advantaged, disadvantaged and neutral mutants) on unstructured meshes is relatively smaller than the corresponding results on regular grids. More importantly, in the case of neutral mutants the introduction of a migration potential has a critical effect on the fixation probability and increases this by orders of magnitude. Further, we examine the effect of boundaries and as intuitively expected, the fixation probability is smaller on the boundary of regular grids when compared to its value in the bulk. Based on these computational results, we speculate on possible better therapeutic strategies that may delay tumor progression to some extent.

Systems mapping of genes controlling chemotherapeutic drug efficiency for cancer stem cells

Cancer can be controlled effectively by using chemotherapeutic drugs to inhibit cancer stem cells, but there is considerable inter-patient variability regarding how these cells respond to drug intervention. Here, we describe a statistical framework for mapping genes that control tumor responses to chemotherapeutic drugs as well as the efficacy of treatments in arresting tumor growth. The framework integrates the mathematical aspects of the cancer stem cell hypothesis into genetic association studies, equipped with a capacity to quantify the magnitude and pattern of genetic effects on the kinetic decline of cancer stem cells in response to therapy. By quantifying how specific genes and their interactions govern drug response, the model provides essential information to tailor personalized drugs for individual patients.

Computer simulation of the metastatic progression

A novel computer model based on a discrete event simulation procedure describes quantitatively the processes underlying the metastatic cascade. Analytical functions describe the size of the primary tumor and the metastases, while a rate function models the intravasation events of the primary tumor and metastases. Events describe the behavior of the malignant cells until the formation of new metastases. The results of the computer simulations are in quantitative agreement with clinical data determined from a patient with hepatocellular carcinoma in the liver. The model provides a more detailed view on the process than a conventional mathematical model. In particular, the implications of interventions on metastasis formation can be calculated.

Clinically relevant modeling of tumor growth and treatment response

Current mathematical models of tumor growth are limited in their clinical application because they require input data that are nearly impossible to obtain with sufficient spatial resolution in patients even at a single time point--for example, extent of vascularization, immune infiltrate, ratio of tumor-to-normal cells, or extracellular matrix status. Here we propose the use of emerging, quantitative tumor imaging methods to initialize a new generation of predictive models. In the near future, these models could be able to forecast clinical outputs, such as overall response to treatment and time to progression, which will provide opportunities for guided intervention and improved patient care.

The development of a fully-integrated immune response model (FIRM) simulator of the immune response through integration of multiple subset models

Background

The complexity and multiscale nature of the mammalian immune response provides an excellent test bed for the potential of mathematical modeling and simulation to facilitate mechanistic understanding. Historically, mathematical models of the immune response focused on subsets of the immune system and/or specific aspects of the response. Mathematical models have been developed for the humoral side of the immune response, or for the cellular side, or for cytokine kinetics, but rarely have they been proposed to encompass the overall system complexity. We propose here a framework for integration of subset models, based on a system biology approach.

Results

A dynamic simulator, the Fully-integrated Immune Response Model (FIRM), was built in a stepwise fashion by integrating published subset models and adding novel features. The approach used to build the model includes the formulation of the network of interacting species and the subsequent introduction of rate laws to describe each biological process. The resulting model represents a multi-organ structure, comprised of the target organ where the immune response takes place, circulating blood, lymphoid T, and lymphoid B tissue. The cell types accounted for include macrophages, a few T-cell lineages (cytotoxic, regulatory, helper 1, and helper 2), and B-cell activation to plasma cells. Four different cytokines were accounted for: IFN-γ, IL-4, IL-10 and IL-12. In addition, generic inflammatory signals are used to represent the kinetics of IL-1, IL-2, and TGF-β. Cell recruitment, differentiation, replication, apoptosis and migration are described as appropriate for the different cell types. The model is a hybrid structure containing information from several mammalian species. The structure of the network was built to be physiologically and biochemically consistent. Rate laws for all the cellular fate processes, growth factor production rates and half-lives, together with antibody production rates and half-lives, are provided. The results demonstrate how this framework can be used to integrate mathematical models of the immune response from several published sources and describe qualitative predictions of global immune system response arising from the integrated, hybrid model. In addition, we show how the model can be expanded to include novel biological findings. Case studies were carried out to simulate TB infection, tumor rejection, response to a blood borne pathogen and the consequences of accounting for regulatory T-cells.

Conclusions

The final result of this work is a postulated and increasingly comprehensive representation of the mammalian immune system, based on physiological knowledge and susceptible to further experimental testing and validation. We believe that the integrated nature of FIRM has the potential to simulate a range of responses under a variety of conditions, from modeling of immune responses after tuberculosis (TB) infection to tumor formation in tissues. FIRM also has the flexibility to be expanded to include both complex and novel immunological response features as our knowledge of the immune system advances.

Microengineered tumor models: insights & opportunities from a physical sciences-oncology perspective

Prevailing evidence has established the fundamental role of microenvironmental conditions in tumorigenesis. However, the ability to identify, interrupt, and translate the underlying cellular and molecular mechanisms into meaningful therapies remains limited, due in part to a lack of organotypic culture systems that accurately recapitulate tumor physiology. Integration of tissue engineering with microfabrication technologies has the potential to address this challenge and mimic tumor heterogeneity with pathological fidelity. Specifically, this approach allows recapitulating global changes of tissue-level phenomena, while also controlling microscale variability of various conditions including spatiotemporal presentation of soluble signals, biochemical and physical characteristics of the extracellular matrix, and cellular composition. Such platforms have continued to elucidate the role of the microenvironment in cancer pathogenesis and significantly improve drug discovery and screening, particularly for therapies that target tumor-enabling stromal components. This review discusses some of the landmark efforts in the field of micro-tumor engineering with a particular emphasis on deregulated tissue organization and mass transport phenomena in the tumor microenvironment.

Dealing with diversity in computational cancer modeling

This paper discusses the need for interconnecting computational cancer models from different sources and scales within clinically relevant scenarios to increase the accuracy of the models and speed up their clinical adaptation, validation, and eventual translation. We briefly review current interoperability efforts drawing upon our experiences with the development of in silico models for predictive oncology within a number of European Commission Virtual Physiological Human initiative projects on cancer. A clinically relevant scenario, addressing brain tumor modeling that illustrates the need for coupling models from different sources and levels of complexity, is described. General approaches to enabling interoperability using XML-based markup languages for biological modeling are reviewed, concluding with a discussion on efforts towards developing cancer-specific XML markup to couple multiple component models for predictive in silico oncology.

Evolution and morphology of microenvironment-enhanced malignancy of three-dimensional invasive solid tumors

The emergence of invasive and metastatic behavior in malignant tumors can often lead to fatal outcomes for patients. The collective malignant tumor behavior resulting from the complex tumor-host interactions and the interactions between the tumor cells is currently poorly understood. In this paper, we employ a cellular automaton (CA) model to investigate microenvironment-enhanced malignant behaviors and morphologies of in vitro avascular invasive solid tumors in three dimensions. Our CA model incorporates a variety of microscopic-scale tumor-host interactions, including the degradation of the extracellular matrix by the malignant cells, nutrient-driven cell migration, pressure buildup due to the deformation of the microenvironment by the growing tumor, and its effect on the local tumor-host interface stability. Moreover, the effects of cell-cell adhesion on tumor growth are explicitly taken into account. Specifically, we find that while strong cell-cell adhesion can suppress the invasive behavior of the tumors growing in soft microenvironments, cancer malignancy can be significantly enhanced by harsh microenvironmental conditions, such as exposure to high pressure levels. We infer from the simulation results a qualitative phase diagram that characterizes the expected malignant behavior of invasive solid tumors in terms of two competing malignancy effects: the rigidity of the microenvironment and cell-cell adhesion. This diagram exhibits phase transitions between noninvasive and invasive behaviors. We also discuss the implications of our results for the diagnosis, prognosis, and treatment of malignant tumors.

From patient-specific mathematical neuro-oncology to precision medicine

Gliomas are notoriously aggressive, malignant brain tumors that have variable response to treatment. These patients often have poor prognosis, informed primarily by histopathology. Mathematical neuro-oncology (MNO) is a young and burgeoning field that leverages mathematical models to predict and quantify response to therapies. These mathematical models can form the basis of modern “precision medicine” approaches to tailor therapy in a patient-specific manner. Patient-specific models (PSMs) can be used to overcome imaging limitations, improve prognostic predictions, stratify patients, and assess treatment response in silico. The information gleaned from such models can aid in the construction and efficacy of clinical trials and treatment protocols, accelerating the pace of clinical research in the war on cancer. This review focuses on the growing translation of PSM to clinical neuro-oncology. It will also provide a forward-looking view on a new era of patient-specific MNO.

An integrated computational/experimental model of lymphoma growth

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

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

A computational model for collective cellular motion in three dimensions: general framework and case study for cell pair dynamics

Cell migration in healthy and diseased systems is a combination of single and collective cell motion. While single cell motion has received considerable attention, our understanding of collective cell motion remains elusive. A new computational framework for the migration of groups of cells in three dimensions is presented, which focuses on the forces acting at the microscopic scale and the interactions between cells and their extracellular matrix (ECM) environment. Cell-cell adhesion, resistance due to the ECM and the factors regulating the propulsion of each cell through the matrix are considered. In particular, our approach emphasizes the role of receptors that mediate cell-cell and cell-matrix interactions, and examines how variation in their properties induces changes in cellular motion. As an important case study, we analyze two interacting cells. Our results show that the dynamics of cell pairs depends on the magnitude and the stochastic nature of the forces. Stronger intercellular stability is generally promoted by surface receptors that move. We also demonstrate that matrix resistance, cellular stiffness and intensity of adhesion contribute to migration behaviors in different ways, with memory effects present that can alter pair motility. If adhesion weakens with time, our findings show that cell pair break-up depends strongly on the way cells interact with the matrix. Finally, the motility for cells in a larger cluster (size 50 cells) is examined to illustrate the full capabilities of the model and to stress the role of cellular pairs in complex cellular structures. Overall, our framework shows how properties of cells and their environment influence the stability and motility of cellular assemblies. This is an important step in the advancement of the understanding of collective motility, and can contribute to knowledge of complex biological processes involving migration, aggregation and detachment of cells in healthy and diseased systems.

Implications of systemic dysfunction for the etiology of malignancy

The current approach to treatment in oncology is to replace the generally cytotoxic chemotherapies with pharmaceutical treatment which inactivates specific molecular targets associated with cancer development and progression. The goal is to limit cellular damage to pathways perceived to be directly responsible for the malignancy. Its underlying assumptions are twofold: (1) that individual pathways are the cause of malignancy; and (2) that the treatment objective should be destruction-either of the tumor or the dysfunctional pathway. However, the extent to which data actually support these assumptions has not been directly addressed. Accumulating evidence suggests that systemic dysfunction precedes the disruption of specific genetic/molecular pathways in most adult cancers and that targeted treatments such as kinase inhibitors may successfully treat one pathway while generating unintended changes to other, non-targeted pathways. This article discusses (1) the systemic basis of malignancy; (2) better profiling of pre-cancerous biomarkers associated with elevated risk so that preventive lifestyle modifications can be instituted early to revert high-risk epigenetic changes before tumors develop; (3) a treatment emphasis in early stage tumors that would target the restoration of systemic balance by strengthening the body's innate defense mechanisms; and (4) establishing better quantitative models of systems to capture adequate complexity for predictability at all stages of tumor progression.

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

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

The effect of interstitial pressure on tumor growth: coupling with the blood and lymphatic vascular systems

The flow of interstitial fluid and the associated interstitial fluid pressure (IFP) in solid tumors and surrounding host tissues have been identified as critical elements in cancer growth and vascularization. Both experimental and theoretical studies have shown that tumors may present elevated IFP, which can be a formidable physical barrier for delivery of cell nutrients and small molecules into the tumor. Elevated IFP may also exacerbate gradients of biochemical signals such as angiogenic factors released by tumors into the surrounding tissues. These studies have helped to understand both biochemical signaling and treatment prognosis. Building upon previous work, here we develop a vascular tumor growth model by coupling a continuous growth model with a discrete angiogenesis model. We include fluid/oxygen extravasation as well as a continuous lymphatic field, and study the micro-environmental fluid dynamics and their effect on tumor growth by accounting for blood flow, transcapillary fluid flux, interstitial fluid flow, and lymphatic drainage. We thus elucidate further the non-trivial relationship between the key elements contributing to the effects of interstitial pressure in solid tumors. In particular, we study the effect of IFP on oxygen extravasation and show that small blood/lymphatic vessel resistance and collapse may contribute to lower transcapillary fluid/oxygen flux, thus decreasing the rate of tumor growth. We also investigate the effect of tumor vascular pathologies, including elevated vascular and interstitial hydraulic conductivities inside the tumor as well as diminished osmotic pressure differences, on the fluid flow across the tumor capillary bed, the lymphatic drainage, and the IFP. Our results reveal that elevated interstitial hydraulic conductivity together with poor lymphatic function is the root cause of the development of plateau profiles of the IFP in the tumor, which have been observed in experiments, and contributes to a more uniform distribution of oxygen, solid tumor pressure and a broad-based collapse of the tumor lymphatics. We also find that the rate that IFF is fluxed into the lymphatics and host tissue is largely controlled by an elevated vascular hydraulic conductivity in the tumor. We discuss the implications of these results on microenvironmental transport barriers, and the tumor invasive and metastatic potential. Our results suggest the possibility of developing strategies of targeting tumor cells based on the cues in the interstitial fluid.

Role of mathematical modeling in bone fracture healing

Bone fracture healing is a complex physiological process commonly described by a four-phase model consisting of an inflammatory phase, two repair phases with soft callus formation followed by hard callus formation, and a remodeling phase, or more recently by an anabolic/catabolic model. Data from humans and animal models have demonstrated crucial environmental conditions for optimal fracture healing, including the mechanical environment, blood supply and availability of mesenchymal stem cells. Fracture healing spans multiple length and time scales, making it difficult to know precisely which factors and/or phases to manipulate in order to obtain optimal fracture-repair outcomes. Deformations resulting from physiological loading or fracture fixation at the organ scale are sensed at the cellular scale by cells inside the fracture callus. These deformations together with autocrine and paracrine signals determine cellular differentiation, proliferation and migration. The local repair activities lead to new bone formation and stabilization of the fracture. Although experimental data are available at different spatial and temporal scales, it is not clear how these data can be linked to provide a holistic view of fracture healing. Mathematical modeling is a powerful tool to quantify conceptual models and to establish the missing links between experimental data obtained at different scales. The objective of this review is to introduce mathematical modeling to readers who are not familiar with this methodology and to demonstrate that once validated, such models can be used for hypothesis testing and to assist in clinical treatment as will be shown for the example of atrophic nonunions.

Analysis and simulation of division- and label-structured population models : a new tool to analyze proliferation assays

In most biological studies and processes, cell proliferation and population dynamics play an essential role. Due to this ubiquity, a multitude of mathematical models has been developed to describe these processes. While the simplest models only consider the size of the overall populations, others take division numbers and labeling of the cells into account. In this work, we present a modeling and computational framework for proliferating cell populations undergoing symmetric cell division, which incorporates both the discrete division number and continuous label dynamics. Thus, it allows for the consideration of division number-dependent parameters as well as the direct comparison of the model prediction with labeling experiments, e.g., performed with Carboxyfluorescein succinimidyl ester (CFSE), and can be shown to be a generalization of most existing models used to describe these data. We prove that under mild assumptions the resulting system of coupled partial differential equations (PDEs) can be decomposed into a system of ordinary differential equations (ODEs) and a set of decoupled PDEs, which drastically reduces the computational effort for simulating the model. Furthermore, the PDEs are solved analytically and the ODE system is truncated, which allows for the prediction of the label distribution of complex systems using a low-dimensional system of ODEs. In addition to modeling the label dynamics, we link the label-induced fluorescence to the measure fluorescence which includes autofluorescence. Furthermore, we provide an analytical approximation for the resulting numerically challenging convolution integral. This is illustrated by modeling and simulating a proliferating population with division number-dependent proliferation rate.

Towards in silico oncology: adapting a four dimensional nephroblastoma treatment model to a clinical trial case based on multi-method sensitivity analysis

In the past decades a great progress in cancer research has been made although medical treatment is still widely based on empirically established protocols which have many limitations. Computational models address such limitations by providing insight into the complex biological mechanisms of tumor progression. A set of clinically-oriented, multiscale models of solid tumor dynamics has been developed by the In Silico Oncology Group (ISOG), Institute of Communication and Computer Systems (ICCS)-National Technical University of Athens (NTUA) to study cancer growth and response to treatment. Within this context using certain representative parameter values, tumor growth and response have been modeled under a cancer preoperative chemotherapy protocol in the framework of the SIOP 2001/GPOH clinical trial. A thorough cross-method sensitivity analysis of the model has been performed. Based on the sensitivity analysis results, a reasonable adaptation of the values of the model parameters to a real clinical case of bilateral nephroblastomatosis has been achieved. The analysis presented supports the potential of the model for the study and eventually the future design of personalized treatment schemes and/or schedules using the data obtained from in vitro experiments and clinical studies.

Understanding immunology via engineering design: the role of mathematical prototyping

A major challenge in immunology is how to translate data into knowledge given the inherent complexity and dynamics of human physiology. Both the physiology and engineering communities have rich histories in applying computational approaches to translate data obtained from complex systems into knowledge of system behavior. However, there are some differences in how disciplines approach problems. By referring to mathematical models as mathematical prototypes, we aim to highlight aspects related to the process (i.e., prototyping) rather than the product (i.e., the model). The objective of this paper is to review how two related engineering concepts, specifically prototyping and "fitness for use," can be applied to overcome the pressing challenge in translating data into improved knowledge of basic immunology that can be used to improve therapies for disease. These concepts are illustrated using two immunology-related examples. The prototypes presented focus on the beta cell mass at the onset of type 1 diabetes and the dynamics of dendritic cells in the lung. This paper is intended to illustrate some of the nuances associated with applying mathematical modeling to improve understanding of the dynamics of disease progression in humans.

In silico modelling of tumour margin diffusion and infiltration: review of current status

As a result of advanced treatment techniques, requiring precise target definitions, a need for more accurate delineation of the Clinical Target Volume (CTV) has arisen. Mathematical modelling is found to be a powerful tool to provide fairly accurate predictions for the Microscopic Extension (ME) of a tumour to be incorporated in a CTV. In general terms, biomathematical models based on a sequence of observations or development of a hypothesis assume some links between biological mechanisms involved in cancer development and progression to provide quantitative or qualitative measures of tumour behaviour as well as tumour response to treatment. Generally, two approaches are taken: deterministic and stochastic modelling. In this paper, recent mathematical models, including deterministic and stochastic methods, are reviewed and critically compared. It is concluded that stochastic models are more promising to provide a realistic description of cancer tumour behaviour due to being intrinsically probabilistic as well as discrete, which enables incorporation of patient-specific biomedical data such as tumour heterogeneity and anatomical boundaries.

Multispecies model of cell lineages and feedback control in solid tumors

We develop a multispecies continuum model to simulate the spatiotemporal dynamics of cell lineages in solid tumors. The model accounts for protein signaling factors produced by cells in lineages, and nutrients supplied by the microenvironment. Together, these regulate the rates of proliferation, self-renewal and differentiation of cells within the lineages, and control cell population sizes and distributions. Terminally differentiated cells release proteins (e.g., from the TGFβ superfamily) that feedback upon less differentiated cells in the lineage both to promote differentiation and decrease rates of proliferation (and self-renewal). Stem cells release a short-range factor that promotes self-renewal (e.g., representative of Wnt signaling factors), as well as a long-range inhibitor of this factor (e.g., representative of Wnt inhibitors such as Dkk and SFRPs). We find that the progression of the tumors and their response to treatment is controlled by the spatiotemporal dynamics of the signaling processes. The model predicts the development of spatiotemporal heterogeneous distributions of the feedback factors (Wnt, Dkk and TGFβ) and tumor cell populations with clusters of stem cells appearing at the tumor boundary, consistent with recent experiments. The nonlinear coupling between the heterogeneous expressions of growth factors and the heterogeneous distributions of cell populations at different lineage stages tends to create asymmetry in tumor shape that may sufficiently alter otherwise homeostatic feedback so as to favor escape from growth control. This occurs in a setting of invasive fingering, and enhanced aggressiveness after standard therapeutic interventions. We find, however, that combination therapy involving differentiation promoters and radiotherapy is very effective in eradicating such a tumor.

Mathematical modeling of human glioma growth based on brain topological structures: study of two clinical cases

Gliomas are the most common primary brain tumors and yet almost incurable due mainly to their great invasion capability. This represents a challenge to present clinical oncology. Here, we introduce a mathematical model aiming to improve tumor spreading capability definition. The model consists in a time dependent reaction-diffusion equation in a three-dimensional spatial domain that distinguishes between different brain topological structures. The model uses a series of digitized images from brain slices covering the whole human brain. The Talairach atlas included in the model describes brain structures at different levels. Also, the inclusion of the Brodmann areas allows prediction of the brain functions affected during tumor evolution and the estimation of correlated symptoms. The model is solved numerically using patient-specific parametrization and finite differences. Simulations consider an initial state with cellular proliferation alone (benign tumor), and an advanced state when infiltration starts (malign tumor). Survival time is estimated on the basis of tumor size and location. The model is used to predict tumor evolution in two clinical cases. In the first case, predictions show that real infiltrative areas are underestimated by current diagnostic imaging. In the second case, tumor spreading predictions were shown to be more accurate than those derived from previous models in the literature. Our results suggest that the inclusion of differential migration in glioma growth models constitutes another step towards a better prediction of tumor infiltration at the moment of surgical or radiosurgical target definition. Also, the addition of physiological/psychological considerations to classical anatomical models will provide a better and integral understanding of the patient disease at the moment of deciding therapeutic options, taking into account not only survival but also life quality.

State of the art in silico tools for the study of signaling pathways in cancer

In the last several years, researchers have exhibited an intense interest in the evolutionarily conserved signaling pathways that have crucial roles during embryonic development. Interestingly, the malfunctioning of these signaling pathways leads to several human diseases, including cancer. The chemical and biophysical events that occur during cellular signaling, as well as the number of interactions within a signaling pathway, make these systems complex to study. In silico resources are tools used to aid the understanding of cellular signaling pathways. Systems approaches have provided a deeper knowledge of diverse biochemical processes, including individual metabolic pathways, signaling networks and genome-scale metabolic networks. In the future, these tools will be enormously valuable, if they continue to be developed in parallel with growing biological knowledge. In this study, an overview of the bioinformatics resources that are currently available for the analysis of biological networks is provided.

Dynamic modeling of genes controlling cancer stem cell proliferation

The growing evidence that cancer originates from stem cells (SC) holds a great promise to eliminate this disease by designing specific drug therapies for removing cancer SC. Translation of this knowledge into predictive tests for the clinic is hampered due to the lack of methods to discriminate cancer SC from non-cancer SC. Here, we address this issue by describing a conceptual strategy for identifying the genetic origins of cancer SC. The strategy incorporates a high-dimensional group of differential equations that characterizes the proliferation, differentiation, and reprogramming of cancer SC in a dynamic cellular and molecular system. The deployment of robust mathematical models will help uncover and explain many still unknown aspects of cell behavior, tissue function, and network organization related to the formation and division of cancer SC. The statistical method developed allows biologically meaningful hypotheses about the genetic control mechanisms of carcinogenesis and metastasis to be tested in a quantitative manner.

Multiscale imaging and computational modeling of blood flow in the tumor vasculature

The evolution in our understanding of tumor angiogenesis has been the result of pioneering imaging and computational modeling studies spanning the endothelial cell, microvasculature and tissue levels. Many of these primary data on the tumor vasculature are in the form of images from pre-clinical tumor models that provide a wealth of qualitative and quantitative information in many dimensions and across different spatial scales. However, until recently, the visualization of changes in the tumor vasculature across spatial scales remained a challenge due to a lack of techniques for integrating micro- and macroscopic imaging data. Furthermore, the paucity of three-dimensional (3-D) tumor vascular data in conjunction with the challenges in obtaining such data from patients presents a serious hurdle for the development and validation of predictive, multiscale computational models of tumor angiogenesis. In this review, we discuss the development of multiscale models of tumor angiogenesis, new imaging techniques capable of reproducing the 3-D tumor vascular architecture with high fidelity, and the emergence of "image-based models" of tumor blood flow and molecular transport. Collectively, these developments are helping us gain a fundamental understanding of the cellular and molecular regulation of tumor angiogenesis that will benefit the development of new cancer therapies. Eventually, we expect this exciting integration of multiscale imaging and mathematical modeling to have widespread application beyond the tumor vasculature to other diseases involving a pathological vasculature, such as stroke and spinal cord injury.

Are metastases from metastases clinical relevant? Computer modelling of cancer spread in a case of hepatocellular carcinoma

BACKGROUND

Metastasis formation remains an enigmatic process and one of the main questions recently asked is whether metastases are able to generate further metastases. Different models have been proposed to answer this question; however, their clinical significance remains unclear. Therefore a computer model was developed that permits comparison of the different models quantitatively with clinical data and that additionally predicts the outcome of treatment interventions.

METHODS

The computer model is based on discrete events simulation approach. On the basis of a case from an untreated patient with hepatocellular carcinoma and its multiple metastases in the liver, it was evaluated whether metastases are able to metastasise and in particular if late disseminated tumour cells are still capable to form metastases. Additionally, the resection of the primary tumour was simulated. The simulation results were compared with clinical data.

RESULTS

The simulation results reveal that the number of metastases varies significantly between scenarios where metastases metastasise and scenarios where they do not. In contrast, the total tumour mass is nearly unaffected by the two different modes of metastasis formation. Furthermore, the results provide evidence that metastasis formation is an early event and that late disseminated tumour cells are still capable of forming metastases. Simulations also allow estimating how the resection of the primary tumour delays the patient's death.

CONCLUSION

The simulation results indicate that for this particular case of a hepatocellular carcinoma late metastases, i.e., metastases from metastases, are irrelevant in terms of total tumour mass. Hence metastases seeded from metastases are clinically irrelevant in our model system. Only the first metastases seeded from the primary tumour contribute significantly to the tumour burden and thus cause the patient's death.

Multiscale cancer modeling

Simulating cancer behavior across multiple biological scales in space and time, i.e., multiscale cancer modeling, is increasingly being recognized as a powerful tool to refine hypotheses, focus experiments, and enable more accurate predictions. A growing number of examples illustrate the value of this approach in providing quantitative insights in the initiation, progression, and treatment of cancer. In this review, we introduce the most recent and important multiscale cancer modeling works that have successfully established a mechanistic link between different biological scales. Biophysical, biochemical, and biomechanical factors are considered in these models. We also discuss innovative, cutting-edge modeling methods that are moving predictive multiscale cancer modeling toward clinical application. Furthermore, because the development of multiscale cancer models requires a new level of collaboration among scientists from a variety of fields such as biology, medicine, physics, mathematics, engineering, and computer science, an innovative Web-based infrastructure is needed to support this growing community.