Integrative mathematical oncology

Translational models of tumor angiogenesis: A nexus of in silico and in vitro models

Emerging evidence shows that endothelial cells are not only the building blocks of vascular networks that enable oxygen and nutrient delivery throughout a tissue but also serve as a rich resource of angiocrine factors. Endothelial cells play key roles in determining cancer progression and response to anti-cancer drugs. Furthermore, the endothelium-specific deposition of extracellular matrix is a key modulator of the availability of angiocrine factors to both stromal and cancer cells. Considering tumor vascular network as a decisive factor in cancer pathogenesis and treatment response, these networks need to be an inseparable component of cancer models. Both computational and in vitro experimental models have been extensively developed to model tumor-endothelium interactions. While informative, they have been developed in different communities and do not yet represent a comprehensive platform. In this review, we overview the necessity of incorporating vascular networks for both in vitro and in silico cancer models and discuss recent progresses and challenges of in vitro experimental microfluidic cancer vasculature-on-chip systems and their in silico counterparts. We further highlight how these two approaches can merge together with the aim of presenting a predictive combinatorial platform for studying cancer pathogenesis and testing the efficacy of single or multi-drug therapeutics for cancer treatment.

The Evolution and Ecology of Resistance in Cancer Therapy

Despite continuous deployment of new treatment strategies and agents over many decades, most disseminated cancers remain fatal. Cancer cells, through their access to the vast information of human genome, have a remarkable capacity to deploy adaptive strategies for even the most effective treatments. We note there are two critical steps in the clinical manifestation of treatment resistance. The first, which is widely investigated, requires deployment of a mechanism of resistance that usually involves increased expression of molecular machinery necessary to eliminate the cytotoxic effect of treatment. However, the emergence of a resistant phenotype is not in itself clinically significant. That is, resistant cells affect patient outcomes only when they form a sufficiently large population to allow tumor progression and treatment failure. Importantly, proliferation of the resistant phenotype is by no means certain and, in fact, depends on complex Darwinian dynamics governed by the costs and benefits of the resistance mechanisms in the context of the local environment and competing populations. Attempts to target molecular machinery of resistance have had little clinical success largely because of the diversity within the human genome-therapeutic interruption of one mechanism simply results in its replacement by an alternative. We explore an alternative strategy for overcoming treatment resistance that seeks to understand and exploit the critical evolutionary dynamics that govern proliferation of the resistant phenotypes. In general, this approach has shown that, although emergence of resistance mechanisms in cancer cells to every current therapy is inevitable, proliferation of the resistant phenotypes is not and can be delayed and even prevented with sufficient understanding of the underlying ecoevolutionary dynamics.

Agent-Based Modeling of Cancer Stem Cell Driven Solid Tumor Growth

Computational modeling of tumor growth has become an invaluable tool to simulate complex cell-cell interactions and emerging population-level dynamics. Agent-based models are commonly used to describe the behavior and interaction of individual cells in different environments. Behavioral rules can be informed and calibrated by in vitro assays, and emerging population-level dynamics may be validated with both in vitro and in vivo experiments. Here, we describe the design and implementation of a lattice-based agent-based model of cancer stem cell driven tumor growth.

Mutation of TGFβ-RII eliminates NSAID cancer chemoprevention

Non-steroidal anti-inflammatory drugs (NSAIDs) exhibit anti-neoplastic (chemoprevention) activity for sporadic cancers and the hereditary cancer predisposition Lynch syndrome (LS/HNPCC). However, the mechanism of NSAID tumor suppression has remained enigmatic. Defects in the core mismatch repair (MMR) genes MSH2 and MLH1 are the principal drivers of LS/HNPCC. Previous work has demonstrated that the villin-Cre^+/−Msh2^flox/flox (VpC-Msh2) mouse is a reliable model for LS/HNPCC intestinal tumorigenesis, which is significantly suppressed by treatment with the NSAID aspirin (ASA) similar to human chemoprevention. Here we show that including a TGFβ receptor type-II (Tgfβ-RII) mutation in the VpC-Msh2 mouse (villin-Cre^+/−Msh2^flox/floxTgfβ−RII^flox/flox) completely eliminates NSAID tumor suppression. These results provide strong genetic evidence that TGFβ signaling and/or effectors participate in NSAID-dependent anti-neoplastic processes and provide fresh avenues for understanding NSAID chemoprevention and resistance.

The role of Allee effect in modelling post resection recurrence of glioblastoma

Resection of the bulk of a tumour often cannot eliminate all cancer cells, due to their infiltration into the surrounding healthy tissue. This may lead to recurrence of the tumour at a later time. We use a reaction-diffusion equation based model of tumour growth to investigate how the invasion front is delayed by resection, and how this depends on the density and behaviour of the remaining cancer cells. We show that the delay time is highly sensitive to qualitative details of the proliferation dynamics of the cancer cell population. The typically assumed logistic type proliferation leads to unrealistic results, predicting immediate recurrence. We find that in glioblastoma cell cultures the cell proliferation rate is an increasing function of the density at small cell densities. Our analysis suggests that cooperative behaviour of cancer cells, analogous to the Allee effect in ecology, can play a critical role in determining the time until tumour recurrence.

Mathematical models of propagating fronts have been used to represent a wide variety of biological phenomena from action potentials in neural cells to invasive species in ecology and epidemic spreading. Here we show that when such models are used to predict the effects of external perturbations the results can be very sensitive to certain details of the local dynamics. For example, the post resection recurrence of tumour growth depends strongly on the density dependence of the proliferation of cancer cells. This suggests that targeting the cooperative behaviour of cancer cells could be an efficient strategy for delaying the recurrence of diffuse aggressive brain tumours.

Multiscale Modeling of Inflammation-Induced Tumorigenesis Reveals Competing Oncogenic and Oncoprotective Roles for Inflammation

Chronic inflammation is a serious risk factor for cancer; however, the routes from inflammation to cancer are poorly understood. On the basis of the processes implicated by frequently mutated genes associated with inflammation and cancer in three organs (stomach, colon, and liver) extracted from the Gene Expression Omnibus, The Cancer Genome Atlas, and Gene Ontology databases, we present a multiscale model of the long-term evolutionary dynamics leading from inflammation to tumorigenesis. The model incorporates cross-talk among interactions on several scales, including responses to DNA damage, gene mutation, cell-cycle behavior, population dynamics, inflammation, and metabolism-immune balance. Model simulations revealed two stages of inflammation-induced tumorigenesis: a precancerous state and tumorigenesis. The precancerous state was mainly caused by mutations in the cell proliferation pathway; the transition from the precancerous to tumorigenic states was induced by mutations in pathways associated with apoptosis, differentiation, and metabolism-immune balance. We identified opposing effects of inflammation on tumorigenesis. Mild inflammation removed cells with DNA damage through DNA damage-induced cell death, whereas severe inflammation accelerated accumulation of mutations and hence promoted tumorigenesis. These results provide insight into the evolutionary dynamics of inflammation-induced tumorigenesis and highlight the combinatorial effects of inflammation and metabolism-immune balance. This approach establishes methods for quantifying cancer risk, for the discovery of driver pathways in inflammation-induced tumorigenesis, and has direct relevance for early detection and prevention and development of new treatment regimes. Cancer Res; 77(22); 6429-41. ©2017 AACR.

A branching process model of heterogeneous DNA damages caused by radiotherapy in in vitro cell cultures

This paper deals with the dynamic modeling and simulation of cell damage heterogeneity and associated mutant cell phenotypes in the therapeutic responses of cancer cell populations submitted to a radiotherapy session during in vitro assays. Each cell is described by a finite number of phenotypic states with possible transitions between them. The population dynamics is then given by an age-dependent multi-type branching process. From this representation, we obtain formulas for the average size of the global survival population as well as the one of subpopulations associated with 10 mutation phenotypes. The proposed model has been implemented into Matlab^© and the numerical results corroborate the ability of the model to reproduce four major types of cell responses: delayed growth, anti-proliferative, cytostatic and cytotoxic.

Homeostasis Back and Forth: An Ecoevolutionary Perspective of Cancer

The role of genetic mutations in cancer is indisputable: They are a key source of tumor heterogeneity and drive its evolution to malignancy. But, the success of these new mutant cells relies on their ability to disrupt the homeostasis that characterizes healthy tissues. Mutated clones unable to break free from intrinsic and extrinsic homeostatic controls will fail to establish a tumor. Here, we will discuss, through the lens of mathematical and computational modeling, why an evolutionary view of cancer needs to be complemented by an ecological perspective to understand why cancer cells invade and subsequently transform their environment during progression. Importantly, this ecological perspective needs to account for tissue homeostasis in the organs that tumors invade, because they perturb the normal regulatory dynamics of these tissues, often coopting them for its own gain. Furthermore, given our current lack of success in treating advanced metastatic cancers through tumor-centric therapeutic strategies, we propose that treatments that aim to restore homeostasis could become a promising venue of clinical research. This ecoevolutionary view of cancer requires mechanistic mathematical models to both integrate clinical with biological data from different scales but also to detangle the dynamic feedback between the tumor and its environment. Importantly, for these models to be useful, they need to embrace a higher degree of complexity than many mathematical modelers are traditionally comfortable with.

Unravelling biology and shifting paradigms in cancer with single-cell sequencing

The fundamental operative unit of a cancer is the genetically and epigenetically innovative single cell. Whether proliferating or quiescent, in the primary tumour mass or disseminated elsewhere, single cells govern the parameters that dictate all facets of the biology of cancer. Thus, single-cell analyses provide the ultimate level of resolution in our quest for a fundamental understanding of this disease. Historically, this quest has been hampered by technological shortcomings. In this Opinion article, we argue that the rapidly evolving field of single-cell sequencing has unshackled the cancer research community of these shortcomings. From furthering an elemental understanding of intra-tumoural genetic heterogeneity and cancer genome evolution to illuminating the governing principles of disease relapse and metastasis, we posit that single-cell sequencing promises to unravel the biology of all facets of this disease.

Fighting Cancer with Mathematics and Viruses

After decades of research, oncolytic virotherapy has recently advanced to clinical application, and currently a multitude of novel agents and combination treatments are being evaluated for cancer therapy. Oncolytic agents preferentially replicate in tumor cells, inducing tumor cell lysis and complex antitumor effects, such as innate and adaptive immune responses and the destruction of tumor vasculature. With the availability of different vector platforms and the potential of both genetic engineering and combination regimens to enhance particular aspects of safety and efficacy, the identification of optimal treatments for patient subpopulations or even individual patients becomes a top priority. Mathematical modeling can provide support in this arena by making use of experimental and clinical data to generate hypotheses about the mechanisms underlying complex biology and, ultimately, predict optimal treatment protocols. Increasingly complex models can be applied to account for therapeutically relevant parameters such as components of the immune system. In this review, we describe current developments in oncolytic virotherapy and mathematical modeling to discuss the benefit of integrating different modeling approaches into biological and clinical experimentation. Conclusively, we propose a mutual combination of these research fields to increase the value of the preclinical development and the therapeutic efficacy of the resulting treatments.

Stochastic model of contact inhibition and the proliferation of melanoma in situ

Contact inhibition is a central feature orchestrating cell proliferation in culture experiments; its loss is associated with malignant transformation and tumorigenesis. We performed a co-culture experiment with human metastatic melanoma cell line (SKMEL- 147) and immortalized keratinocyte cells (HaCaT). After 8 days a spatial pattern was detected, characterized by the formation of clusters of melanoma cells surrounded by keratinocytes constraining their proliferation. In addition, we observed that the proportion of melanoma cells within the total population has increased. To explain our results we propose a spatial stochastic model (following a philosophy of the Widom-Rowlinson model from Statistical Physics and Molecular Chemistry) which considers cell proliferation, death, migration, and cell-to-cell interaction through contact inhibition. Our numerical simulations demonstrate that loss of contact inhibition is a sufficient mechanism, appropriate for an explanation of the increase in the proportion of tumor cells and generation of spatial patterns established in the conducted experiments.

Predicting Patient-Specific Radiotherapy Protocols Based on Mathematical Model Choice for Proliferation Saturation Index

Radiation is commonly used in cancer treatment. Over 50% of all cancer patients will undergo radiotherapy (RT) as part of cancer care. Scientific advances in RT have primarily focused on the physical characteristics of treatment including beam quality and delivery. Only recently have inroads been made into utilizing tumor biology and radiobiology to design more appropriate RT protocols. Tumors are composites of proliferating and growth-arrested cells, and overall response depends on their respective proportions at irradiation. Prokopiou et al. (Radiat Oncol 10:159, 2015) developed the concept of the proliferation saturation index (PSI) to augment the clinical decision process associated with RT. This framework is based on the application of the logistic equation to pre-treatment imaging data in order to estimate a patient-specific tumor carrying capacity, which is then used to recommend a specific RT protocol. It is unclear, however, how dependent clinical recommendations are on the underlying tumor growth law. We discuss a PSI framework with a generalized logistic equation that can capture kinetics of different well-known growth laws including logistic and Gompertzian growth. Estimation of model parameters on the basis of clinical data revealed that the generalized logistic model can describe data equally well for a wide range of the generalized logistic exponent value. Clinical recommendations based on the calculated PSI, however, are strongly dependent on the specific growth law assumed. Our analysis suggests that the PSI framework may best be utilized in clinical practice when the underlying tumor growth law is known, or when sufficiently many tumor growth models suggest similar fractionation protocols.

Epithelial/mesenchymal plasticity: how have quantitative mathematical models helped improve our understanding?

Phenotypic plasticity, the ability of cells to reversibly alter their phenotypes in response to signals, presents a significant clinical challenge to treating solid tumors. Tumor cells utilize phenotypic plasticity to evade therapies, metastasize, and colonize distant organs. As a result, phenotypic plasticity can accelerate tumor progression. A well‐studied example of phenotypic plasticity is the bidirectional conversions among epithelial, mesenchymal, and hybrid epithelial/mesenchymal (E/M) phenotype(s). These conversions can alter a repertoire of cellular traits associated with multiple hallmarks of cancer, such as metabolism, immune evasion, invasion, and metastasis. To tackle the complexity and heterogeneity of these transitions, mathematical models have been developed that seek to capture the experimentally verified molecular mechanisms and act as ‘hypothesis‐generating machines’. Here, we discuss how these quantitative mathematical models have helped us explain existing experimental data, guided further experiments, and provided an improved conceptual framework for understanding how multiple intracellular and extracellular signals can drive E/M plasticity at both the single‐cell and population levels. We also discuss the implications of this plasticity in driving multiple aggressive facets of tumor progression.

Successive range expansion promotes diversity and accelerates evolution in spatially structured microbial populations

Successive range expansions occur within all domains of life, where one population expands first (primary expansion) and one or more secondary populations then follow (secondary expansion). In general, genetic drift reduces diversity during range expansion. However, it is not clear whether the same effect applies during successive range expansion, mainly because the secondary population must expand into space occupied by the primary population. Here we used an experimental microbial model system to show that, in contrast to primary range expansion, successive range expansion promotes local population diversity. Because of mechanical constraints imposed by the presence of the primary population, the secondary population forms fractal-like dendritic structures. This divides the advancing secondary population into many small sub-populations and promotes intermixing between the primary and secondary populations. We further developed a mathematical model to simulate the formation of dendritic structures in the secondary population during succession. By introducing mutations in the primary or dendritic secondary populations, we found that mutations are more likely to accumulate in the dendritic secondary populations. Our results thus show that successive range expansion can promote intermixing over the short term and increase genetic diversity over the long term. Our results therefore have potentially important implications for predicting the ecological processes and evolutionary trajectories of microbial communities.

The future of personalised radiotherapy for head and neck cancer

Radiotherapy has long been the mainstay of treatment for patients with head and neck cancer and has traditionally involved a stage-dependent strategy whereby all patients with the same TNM stage receive the same therapy. We believe there is a substantial opportunity to improve radiotherapy delivery beyond just technological and anatomical precision. In this Series paper, we explore several new ideas that could improve understanding of the phenotypic and genotypic differences that exist between patients and their tumours. We discuss how exploiting these differences and taking advantage of precision medicine tools-such as genomics, radiomics, and mathematical modelling-could open new doors to personalised radiotherapy adaptation and treatment. We propose a new treatment shift that moves away from an era of empirical dosing and fractionation to an era focused on the development of evidence to guide personalisation and biological adaptation of radiotherapy. We believe these approaches offer the potential to improve outcomes and reduce toxicity.

Systems Chronotherapeutics

Chronotherapeutics aim at treating illnesses according to the endogenous biologic rhythms, which moderate xenobiotic metabolism and cellular drug response. The molecular clocks present in individual cells involve approximately fifteen clock genes interconnected in regulatory feedback loops. They are coordinated by the suprachiasmatic nuclei, a hypothalamic pacemaker, which also adjusts the circadian rhythms to environmental cycles. As a result, many mechanisms of diseases and drug effects are controlled by the circadian timing system. Thus, the tolerability of nearly 500 medications varies by up to fivefold according to circadian scheduling, both in experimental models and/or patients. Moreover, treatment itself disrupted, maintained, or improved the circadian timing system as a function of drug timing. Improved patient outcomes on circadian-based treatments (chronotherapy) have been demonstrated in randomized clinical trials, especially for cancer and inflammatory diseases. However, recent technological advances have highlighted large interpatient differences in circadian functions resulting in significant variability in chronotherapy response. Such findings advocate for the advancement of personalized chronotherapeutics through interdisciplinary systems approaches. Thus, the combination of mathematical, statistical, technological, experimental, and clinical expertise is now shaping the development of dedicated devices and diagnostic and delivery algorithms enabling treatment individualization. In particular, multiscale systems chronopharmacology approaches currently combine mathematical modeling based on cellular and whole-body physiology to preclinical and clinical investigations toward the design of patient-tailored chronotherapies. We review recent systems research works aiming to the individualization of disease treatment, with emphasis on both cancer management and circadian timing system-resetting strategies for improving chronic disease control and patient outcomes.

Catch my drift? Making sense of genomic intra-tumour heterogeneity

The cancer genome is shaped by three components of the evolutionary process: mutation, selection and drift. While many studies have focused on the first two components, the role of drift in cancer evolution has received little attention. Drift occurs when all individuals in the population have the same likelihood of producing surviving offspring, and so by definition a drifting population is one that is evolving neutrally. Here we focus on how neutral evolution is manifested in the cancer genome. We discuss how neutral passenger mutations provide a magnifying glass that reveals the evolutionary dynamics underpinning cancer development, and outline how statistical inference can be used to quantify these dynamics from sequencing data. We argue that only after we understand the impact of neutral drift on the genome can we begin to make full sense of clonal selection. This article is part of a Special Issue entitled: Evolutionary principles - heterogeneity in cancer? Edited by Dr. Robert A. Gatenby.

Therapeutic Potential of Bacteria against Solid Tumors

Intentional bacterial infections can produce efficacious antitumor responses in mice, rats, dogs, and humans. However, low overall success rates and intense side effects prevent such approaches from being employed clinically. In this work, we titered bacteria and/or the proinflammatory cytokine TNFα in a set of established murine models of cancer. To interpret the experiments conducted, we considered and calibrated a tumor-effector cell recruitment model under the influence of functional tumor-associated vasculature. In this model, bacterial infections and TNFα enhanced immune activity and altered vascularization in the tumor bed. Information to predict bacterial therapy outcomes was provided by pretreatment tumor size and the underlying immune recruitment dynamics. Notably, increasing bacterial loads did not necessarily produce better long-term tumor control, suggesting that tumor sizes affected optimal bacterial loads. Short-term treatment responses were favored by high concentrations of effector cells postinjection, such as induced by higher bacterial loads, but in the longer term did not correlate with an effective restoration of immune surveillance. Overall, our findings suggested that a combination of intermediate bacterial loads with low levels TNFα administration could enable more favorable outcomes elicited by bacterial infections in tumor-bearing subjects. Cancer Res; 77(7); 1553-63. ©2017 AACR.

Mechanistic mathematical models: An underused platform for HPV research

Health economic modeling has become an invaluable methodology for the design and evaluation of clinical and public health interventions against the human papillomavirus (HPV) and associated diseases. At the same time, relatively little attention has been paid to a different yet complementary class of models, namely that of mechanistic mathematical models. The primary focus of mechanistic mathematical models is to better understand the intricate biologic mechanisms and dynamics of disease. Inspired by a long and successful history of mechanistic modeling in other biomedical fields, we highlight several areas of HPV research where mechanistic models have the potential to advance the field. We argue that by building quantitative bridges between biologic mechanism and population level data, mechanistic mathematical models provide a unique platform to enable collaborations between experimentalists who collect data at different physical scales of the HPV infection process. Through such collaborations, mechanistic mathematical models can accelerate and enhance the investigation of HPV and related diseases.

Three-dimensional Image-based Mechanical Modeling for Predicting the Response of Breast Cancer to Neoadjuvant Therapy

The use of quantitative medical imaging data to initialize and constrain mechanistic mathematical models of tumor growth has demonstrated a compelling strategy for predicting therapeutic response. More specifically, we have demonstrated a data-driven framework for prediction of residual tumor burden following neoadjuvant therapy in breast cancer that uses a biophysical mathematical model combining reaction-diffusion growth/therapy dynamics and biomechanical effects driven by early time point imaging data. Whereas early work had been based on a limited dimensionality reduction (two-dimensional planar modeling analysis) to simplify the numerical implementation, in this work, we extend our framework to a fully volumetric, three-dimensional biophysical mathematical modeling approach in which parameter estimates are generated by an inverse problem based on the adjoint state method for numerical efficiency. In an in silico performance study, we show accurate parameter estimation with error less than 3% as compared to ground truth. We apply the approach to patient data from a patient with pathological complete response and a patient with residual tumor burden and demonstrate technical feasibility and predictive potential with direct comparisons between imaging data observation and model predictions of tumor cellularity and volume. Comparisons to our previous two-dimensional modeling framework reflect enhanced model prediction of residual tumor burden through the inclusion of additional imaging slices of patient-specific data.

Parallelization and High-Performance Computing Enables Automated Statistical Inference of Multi-scale Models

Mechanistic understanding of multi-scale biological processes, such as cell proliferation in a changing biological tissue, is readily facilitated by computational models. While tools exist to construct and simulate multi-scale models, the statistical inference of the unknown model parameters remains an open problem. Here, we present and benchmark a parallel approximate Bayesian computation sequential Monte Carlo (pABC SMC) algorithm, tailored for high-performance computing clusters. pABC SMC is fully automated and returns reliable parameter estimates and confidence intervals. By running the pABC SMC algorithm for ∼10⁶ hr, we parameterize multi-scale models that accurately describe quantitative growth curves and histological data obtained in vivo from individual tumor spheroid growth in media droplets. The models capture the hybrid deterministic-stochastic behaviors of 10⁵-10⁶ of cells growing in a 3D dynamically changing nutrient environment. The pABC SMC algorithm reliably converges to a consistent set of parameters. Our study demonstrates a proof of principle for robust, data-driven modeling of multi-scale biological systems and the feasibility of multi-scale model parameterization through statistical inference.

Why one-size-fits-all vaso-modulatory interventions fail to control glioma invasion: in silico insights

Gliomas are highly invasive brain tumours characterised by poor prognosis and limited response to therapy. There is an ongoing debate on the therapeutic potential of vaso-modulatory interventions against glioma invasion. Prominent vasculature-targeting therapies involve tumour blood vessel deterioration and normalisation. The former aims at tumour infarction and nutrient deprivation induced by blood vessel occlusion/collapse. In contrast, the therapeutic intention of normalising the abnormal tumour vasculature is to improve the efficacy of conventional treatment modalities. Although these strategies have shown therapeutic potential, it remains unclear why they both often fail to control glioma growth. To shed some light on this issue, we propose a mathematical model based on the migration/proliferation dichotomy of glioma cells in order to investigate why vaso-modulatory interventions have shown limited success in terms of tumour clearance. We found the existence of a critical cell proliferation/diffusion ratio that separates glioma responses to vaso-modulatory interventions into two distinct regimes. While for tumours, belonging to one regime, vascular modulations reduce the front speed and increase the infiltration width, for those in the other regime, the invasion speed increases and infiltration width decreases. We discuss how these in silico findings can be used to guide individualised vaso-modulatory approaches to improve treatment success rates.

Tissue-scale, personalized modeling and simulation of prostate cancer growth

Recently, mathematical modeling and simulation of diseases and their treatments have enabled the prediction of clinical outcomes and the design of optimal therapies on a personalized (i.e., patient-specific) basis. This new trend in medical research has been termed "predictive medicine." Prostate cancer (PCa) is a major health problem and an ideal candidate to explore tissue-scale, personalized modeling of cancer growth for two main reasons: First, it is a small organ, and, second, tumor growth can be estimated by measuring serum prostate-specific antigen (PSA, a PCa biomarker in blood), which may enable in vivo validation. In this paper, we present a simple continuous model that reproduces the growth patterns of PCa. We use the phase-field method to account for the transformation of healthy cells to cancer cells and use diffusion-reaction equations to compute nutrient consumption and PSA production. To accurately and efficiently compute tumor growth, our simulations leverage isogeometric analysis (IGA). Our model is shown to reproduce a known shape instability from a spheroidal pattern to fingered growth. Results of our computations indicate that such shift is a tumor response to escape starvation, hypoxia, and, eventually, necrosis. Thus, branching enables the tumor to minimize the distance from inner cells to external nutrients, contributing to cancer survival and further development. We have also used our model to perform tissue-scale, personalized simulation of a PCa patient, based on prostatic anatomy extracted from computed tomography images. This simulation shows tumor progression similar to that seen in clinical practice.

Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis

We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.

Coupled Hybrid Continuum-Discrete Model of Tumor Angiogenesis and Growth

The processes governing tumor growth and angiogenesis are codependent. To study the relationship between them, we proposed a coupled hybrid continuum-discrete model. In this model, tumor cells, their microenvironment (extracellular matrixes, matrix-degrading enzymes, and tumor angiogenic factors), and their network of blood vessels, described by a series of discrete points, were considered. The results of numerical simulation reveal the process of tumor growth and the change in microenvironment from avascular to vascular stage, indicating that the network of blood vessels develops gradually as the tumor grows. Our findings also reveal that a tumor is divided into three regions: necrotic, semi-necrotic, and well-vascularized. The results agree well with the previous relevant studies and physiological facts, and this model represents a platform for further investigations of tumor therapy.

Tumor Static Concentration Curves in Combination Therapy

Combination therapies are widely accepted as a cornerstone for treatment of different cancer types. A tumor growth inhibition (TGI) model is developed for combinations of cetuximab and cisplatin obtained from xenograft mice. Unlike traditional TGI models, both natural cell growth and cell death are considered explicitly. The growth rate was estimated to 0.006 h^-1 and the natural cell death to 0.0039 h^-1 resulting in a tumor doubling time of 14 days. The tumor static concentrations (TSC) are predicted for each individual compound. When the compounds are given as single-agents, the required concentrations were computed to be 506 μg · mL^-1 and 56 ng · mL^-1 for cetuximab and cisplatin, respectively. A TSC curve is constructed for different combinations of the two drugs, which separates concentration combinations into regions of tumor shrinkage and tumor growth. The more concave the TSC curve is, the lower is the total exposure to test compounds necessary to achieve tumor regression. The TSC curve for cetuximab and cisplatin showed weak concavity. TSC values and TSC curves were estimated that predict tumor regression for 95% of the population by taking between-subject variability into account. The TSC concept is further discussed for different concentration-effect relationships and for combinations of three or more compounds.

The tumor as an organ: comprehensive spatial and temporal modeling of the tumor and its microenvironment

BACKGROUND

Research related to cancer is vast, and continues in earnest in many directions. Due to the complexity of cancer, a better understanding of tumor growth dynamics can be gleaned from a dynamic computational model. We present a comprehensive, fully executable, spatial and temporal 3D computational model of the development of a cancerous tumor together with its environment.

RESULTS

The model was created using Statecharts, which were then connected to an interactive animation front-end that we developed especially for this work, making it possible to visualize on the fly the on-going events of the system's execution, as well as the effect of various input parameters. We were thus able to gain a better understanding of, e.g., how different amounts or thresholds of oxygen and VEGF (vascular endothelial growth factor) affect the progression of the tumor. We found that the tumor has a critical turning point, where it either dies or recovers. If minimum conditions are met at that time, it eventually develops into a full, active, growing tumor, regardless of the actual amount; otherwise it dies.

CONCLUSIONS

This brings us to the conclusion that the tumor is in fact a very robust system: changing initial values of VEGF and oxygen can increase the time it takes to become fully developed, but will not necessarily completely eliminate it.

Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations

Background

A thorough understanding of the ecological and evolutionary mechanisms that drive the phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer research community. In this respect, mathematical modelling can complement experimental cancer research by offering alternative means of understanding the results of in vitro and in vivo experiments, and by allowing for a quick and easy exploration of a variety of biological scenarios through in silico studies.

Results

To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here a phenotype-structured model of evolutionary dynamics in a cancer cell population which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size of the cell population are shaped by the strength of natural selection, the rate of random epimutations, the intensity of the competition for limited resources between cells, and the drug dose in use.

Conclusions

Our analytical results clarify the conditions for the successful adaptation of cancer cells faced with environmental changes. Furthermore, the results of our analyses demonstrate that the same cell population exposed to different concentrations of the same cytotoxic drug can take different evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents is more complex than a simple binary outcome, i.e., extinction of sensitive cells and selection of highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents at high doses can act as a double-edged sword by promoting the outgrowth of drug resistant cellular clones. Overall, our theoretical work offers a formal basis for the development of anti-cancer therapeutic protocols that go beyond the ‘maximum-tolerated-dose paradigm’, as they may be more effective than traditional protocols at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones.

Reviewers

This article was reviewed by Angela Pisco, Sébastien Benzekry and Heiko Enderling.

Electronic supplementary material

The online version of this article (doi:10.1186/s13062-016-0143-4) contains supplementary material, which is available to authorized users.

Predictive computational modeling to define effective treatment strategies for bone metastatic prostate cancer

The ability to rapidly assess the efficacy of therapeutic strategies for incurable bone metastatic prostate cancer is an urgent need. Pre-clinical in vivo models are limited in their ability to define the temporal effects of therapies on simultaneous multicellular interactions in the cancer-bone microenvironment. Integrating biological and computational modeling approaches can overcome this limitation. Here, we generated a biologically driven discrete hybrid cellular automaton (HCA) model of bone metastatic prostate cancer to identify the optimal therapeutic window for putative targeted therapies. As proof of principle, we focused on TGFβ because of its known pleiotropic cellular effects. HCA simulations predict an optimal effect for TGFβ inhibition in a pre-metastatic setting with quantitative outputs indicating a significant impact on prostate cancer cell viability, osteoclast formation and osteoblast differentiation. In silico predictions were validated in vivo with models of bone metastatic prostate cancer (PAIII and C4-2B). Analysis of human bone metastatic prostate cancer specimens reveals heterogeneous cancer cell use of TGFβ. Patient specific information was seeded into the HCA model to predict the effect of TGFβ inhibitor treatment on disease evolution. Collectively, we demonstrate how an integrated computational/biological approach can rapidly optimize the efficacy of potential targeted therapies on bone metastatic prostate cancer.

Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation

BACKGROUND

Drug-induced drug resistance in cancer has been attributed to diverse biological mechanisms at the individual cell or cell population scale, relying on stochastically or epigenetically varying expression of phenotypes at the single cell level, and on the adaptability of tumours at the cell population level.

SCOPE OF REVIEW

We focus on intra-tumour heterogeneity, namely between-cell variability within cancer cell populations, to account for drug resistance. To shed light on such heterogeneity, we review evolutionary mechanisms that encompass the great evolution that has designed multicellular organisms, as well as smaller windows of evolution on the time scale of human disease. We also present mathematical models used to predict drug resistance in cancer and optimal control methods that can circumvent it in combined therapeutic strategies.

MAJOR CONCLUSIONS

Plasticity in cancer cells, i.e., partial reversal to a stem-like status in individual cells and resulting adaptability of cancer cell populations, may be viewed as backward evolution making cancer cell populations resistant to drug insult. This reversible plasticity is captured by mathematical models that incorporate between-cell heterogeneity through continuous phenotypic variables. Such models have the benefit of being compatible with optimal control methods for the design of optimised therapeutic protocols involving combinations of cytotoxic and cytostatic treatments with epigenetic drugs and immunotherapies.

GENERAL SIGNIFICANCE

Gathering knowledge from cancer and evolutionary biology with physiologically based mathematical models of cell population dynamics should provide oncologists with a rationale to design optimised therapeutic strategies to circumvent drug resistance, that still remains a major pitfall of cancer therapeutics. This article is part of a Special Issue entitled "System Genetics" Guest Editor: Dr. Yudong Cai and Dr. Tao Huang.

Identification of Novel Pathways in Plant Lectin-Induced Cancer Cell Apoptosis

Plant lectins have been investigated to elucidate their complicated mechanisms due to their remarkable anticancer activities. Although plant lectins seems promising as a potential anticancer agent for further preclinical and clinical uses, further research is still urgently needed and should include more focus on molecular mechanisms. Herein, a Naïve Bayesian model was developed to predict the protein-protein interaction (PPI), and thus construct the global human PPI network. Moreover, multiple sources of biological data, such as smallest shared biological process (SSBP), domain-domain interaction (DDI), gene co-expression profiles and cross-species interolog mapping were integrated to build the core apoptotic PPI network. In addition, we further modified it into a plant lectin-induced apoptotic cell death context. Then, we identified 22 apoptotic hub proteins in mesothelioma cells according to their different microarray expressions. Subsequently, we used combinational methods to predict microRNAs (miRNAs) which could negatively regulate the abovementioned hub proteins. Together, we demonstrated the ability of our Naïve Bayesian model-based network for identifying novel plant lectin-treated cancer cell apoptotic pathways. These findings may provide new clues concerning plant lectins as potential apoptotic inducers for cancer drug discovery.

Integrating Multiscale Modeling with Drug Effects for Cancer Treatment

In this paper, we review multiscale modeling for cancer treatment with the incorporation of drug effects from an applied system's pharmacology perspective. Both the classical pharmacology and systems biology are inherently quantitative; however, systems biology focuses more on networks and multi factorial controls over biological processes rather than on drugs and targets in isolation, whereas systems pharmacology has a strong focus on studying drugs with regard to the pharmacokinetic (PK) and pharmacodynamic (PD) relations accompanying drug interactions with multiscale physiology as well as the prediction of dosage-exposure responses and economic potentials of drugs. Thus, it requires multiscale methods to address the need for integrating models from the molecular levels to the cellular, tissue, and organism levels. It is a common belief that tumorigenesis and tumor growth can be best understood and tackled by employing and integrating a multifaceted approach that includes in vivo and in vitro experiments, in silico models, multiscale tumor modeling, continuous/discrete modeling, agent-based modeling, and multiscale modeling with PK/PD drug effect inputs. We provide an example application of multiscale modeling employing stochastic hybrid system for a colon cancer cell line HCT-116 with the application of Lapatinib drug. It is observed that the simulation results are similar to those observed from the setup of the wet-lab experiments at the Translational Genomics Research Institute.

Simulation of the Protein-Shedding Kinetics of a Fully Vascularized Tumor

Circulating biomarkers are of significant interest for cancer detection and treatment personalization. However, the biophysical processes that determine how proteins are shed from cancer cells or their microenvironment, diffuse through tissue, enter blood vasculature, and persist in circulation remain poorly understood. Since approaches primarily focused on experimental evaluation are incapable of measuring the shedding and persistence for every possible marker candidate, we propose an interdisciplinary computational/experimental approach that includes computational modeling of tumor tissue heterogeneity. The model implements protein production, transport, and shedding based on tumor vascularization, cell proliferation, hypoxia, and necrosis, thus quantitatively relating the tumor and circulating proteomes. The results highlight the dynamics of shedding as a function of protein diffusivity and production. Linking the simulated tumor parameters to clinical tumor and vascularization measurements could potentially enable this approach to reveal the tumor-specific conditions based on the protein detected in circulation and thus help to more accurately manage cancer diagnosis and treatment.

The mathematics of cancer: integrating quantitative models

Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intra-tumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.

Mathematical model and its fast numerical method for the tumor growth

In this paper, we reformulate the diffuse interface model of the tumor growth (S.M. Wise et al., Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524--543). In the new proposed model, we use the conservative second-order Allen--Cahn equation with a space--time dependent Lagrange multiplier instead of using the fourth-order Cahn--Hilliard equation in the original model. To numerically solve the new model, we apply a recently developed hybrid numerical method. We perform various numerical experiments. The computational results demonstrate that the new model is not only fast but also has a good feature such as distributing excess mass from the inside of tumor to its boundary regions.

Current Challenges in Glioblastoma: Intratumour Heterogeneity, Residual Disease, and Models to Predict Disease Recurrence

Glioblastoma (GB) is the most common primary malignant brain tumor, and despite the availability of chemotherapy and radiotherapy to combat the disease, overall survival remains low with a high incidence of tumor recurrence. Technological advances are continually improving our understanding of the disease, and in particular, our knowledge of clonal evolution, intratumor heterogeneity, and possible reservoirs of residual disease. These may inform how we approach clinical treatment and recurrence in GB. Mathematical modeling (including neural networks) and strategies such as multiple sampling during tumor resection and genetic analysis of circulating cancer cells, may be of great future benefit to help predict the nature of residual disease and resistance to standard and molecular therapies in GB.

Application of Evolutionary Principles to Cancer Therapy

The dynamic cancer ecosystem, with its rich temporal and spatial diversity in environmental conditions and heritable cell phenotypes, is remarkably robust to therapeutic perturbations. Even when response to therapy is clinically complete, adaptive tumor strategies almost inevitably emerge and the tumor returns. Although evolution of resistance remains the proximate cause of death in most cancer patients, a recent analysis found that evolutionary terms were included in less than 1% of articles on the cancer treatment outcomes, and this has not changed in 30 years. Here, we review treatment methods that attempt to understand and exploit intratumoral evolution to prolong response to therapy. In general, we find that treating metastatic (i.e., noncurable) cancers using the traditional strategy aimed at killing the maximum number of tumor cells is evolutionarily unsound because, by eliminating all treatment-sensitive cells, it enables rapid proliferation of resistant populations-a well-known evolutionary phenomenon termed "competitive release." Alternative strategies, such as adaptive therapy, "ersatzdroges," and double-bind treatments, shift focus from eliminating tumor cells to evolution-based methods that suppress growth of resistant populations to maintain long-term control.

Modeling Invasion Dynamics with Spatial Random-Fitness Due to Micro-Environment

Numerous experimental studies have demonstrated that the microenvironment is a key regulator influencing the proliferative and migrative potentials of species. Spatial and temporal disturbances lead to adverse and hazardous microenvironments for cellular systems that is reflected in the phenotypic heterogeneity within the system. In this paper, we study the effect of microenvironment on the invasive capability of species, or mutants, on structured grids (in particular, square lattices) under the influence of site-dependent random proliferation in addition to a migration potential. We discuss both continuous and discrete fitness distributions. Our results suggest that the invasion probability is negatively correlated with the variance of fitness distribution of mutants (for both advantageous and neutral mutants) in the absence of migration of both types of cells. A similar behaviour is observed even in the presence of a random fitness distribution of host cells in the system with neutral fitness rate. In the case of a bimodal distribution, we observe zero invasion probability until the system reaches a (specific) proportion of advantageous phenotypes. Also, we find that the migrative potential amplifies the invasion probability as the variance of fitness of mutants increases in the system, which is the exact opposite in the absence of migration. Our computational framework captures the harsh microenvironmental conditions through quenched random fitness distributions and migration of cells, and our analysis shows that they play an important role in the invasion dynamics of several biological systems such as bacterial micro-habitats, epithelial dysplasia, and metastasis. We believe that our results may lead to more experimental studies, which can in turn provide further insights into the role and impact of heterogeneous environments on invasion dynamics.

An Emerging Allee Effect Is Critical for Tumor Initiation and Persistence

Tumor cells develop different strategies to cope with changing microenvironmental conditions. A prominent example is the adaptive phenotypic switching between cell migration and proliferation. While it has been shown that the migration-proliferation plasticity influences tumor spread, it remains unclear how this particular phenotypic plasticity affects overall tumor growth, in particular initiation and persistence. To address this problem, we formulate and study a mathematical model of spatio-temporal tumor dynamics which incorporates the microenvironmental influence through a local cell density dependence. Our analysis reveals that two dynamic regimes can be distinguished. If cell motility is allowed to increase with local cell density, any tumor cell population will persist in time, irrespective of its initial size. On the contrary, if cell motility is assumed to decrease with respect to local cell density, any tumor population below a certain size threshold will eventually extinguish, a fact usually termed as Allee effect in ecology. These results suggest that strategies aimed at modulating migration are worth to be explored as alternatives to those mainly focused at keeping tumor proliferation under control.

Controlling tumor growth remains a major medical challenge. Current clinical therapies focus on strategies to reduce tumor cell proliferation. However, during tumor progression, tumor cells may switch between proliferative and migratory behaviors, thereby allowing adaptation to microenvironmental changes that result in variations in local tumor cell density. We herein explore by means of a mathematical model the impact of migration-proliferation plasticity on tumor initiation and persistence. Our work suggests that small tumors can become extinct solely by their intrinsic cell population dynamics if cell motility decreases along with local cell density. In contrast, if cell motility increases with cell density, the tumor inevitably grows. Our model suggests that the regulation of cell migration plays a key role in tumor growth as a whole, making this feature a potential target for clinical studies.

Evolutionary determinants of cancer

Our understanding of cancer is being transformed by exploring clonal diversity, drug resistance, and causation within an evolutionary framework. The therapeutic resilience of advanced cancer is a consequence of its character as a complex, dynamic, and adaptive ecosystem engendering robustness, underpinned by genetic diversity and epigenetic plasticity. The risk of mutation-driven escape by self-renewing cells is intrinsic to multicellularity but is countered by multiple restraints, facilitating increasing complexity and longevity of species. But our own species has disrupted this historical narrative by rapidly escalating intrinsic risk. Evolutionary principles illuminate these challenges and provide new avenues to explore for more effective control.

SIGNIFICANCE

Lifetime risk of cancer now approximates to 50% in Western societies. And, despite many advances, the outcome for patients with disseminated disease remains poor, with drug resistance the norm. An evolutionary perspective may provide a clearer understanding of how cancer clones develop robustness and why, for us as a species, risk is now off the scale. And, perhaps, of what we might best do to achieve more effective control.

Evolutionary determinants of cancer

Our understanding of cancer is being transformed by exploring clonal diversity, drug resistance, and causation within an evolutionary framework. The therapeutic resilience of advanced cancer is a consequence of its character as a complex, dynamic, and adaptive ecosystem engendering robustness, underpinned by genetic diversity and epigenetic plasticity. The risk of mutation-driven escape by self-renewing cells is intrinsic to multicellularity but is countered by multiple restraints, facilitating increasing complexity and longevity of species. But our own species has disrupted this historical narrative by rapidly escalating intrinsic risk. Evolutionary principles illuminate these challenges and provide new avenues to explore for more effective control.

SIGNIFICANCE

Lifetime risk of cancer now approximates to 50% in Western societies. And, despite many advances, the outcome for patients with disseminated disease remains poor, with drug resistance the norm. An evolutionary perspective may provide a clearer understanding of how cancer clones develop robustness and why, for us as a species, risk is now off the scale. And, perhaps, of what we might best do to achieve more effective control.

An Interactive Tool for Animating Biology, and Its Use in Spatial and Temporal Modeling of a Cancerous Tumor and Its Microenvironment

The ability to visualize the ongoing events of a computational model of biology is critical, both in order to see the dynamics of the biological system in action and to enable interaction with the model from which one can observe the resulting behavior. To this end, we have built a new interactive animation tool, SimuLife, for visualizing reactive models of cellular biology. SimuLife is web-based, and is freely accessible at http://simulife.weizmann.ac.il/. We have used SimuLife to animate a model that describes the development of a cancerous tumor, based on the individual components of the system and its environment. This has helped in understanding the dynamics of the tumor and its surrounding blood vessels, and in verifying the behavior, fine-tuning the model accordingly, and learning in which way different factors affect the tumor.

Clinical management of breast cancer heterogeneity

Traditionally, intertumour heterogeneity in breast cancer has been documented in terms of different histological subtypes, treatment sensitivity profiles, and clinical outcomes among different patients. Results of high-throughput molecular profiling studies have subsequently revealed the true extent of this heterogeneity. Further complicating this scenario, the heterogeneous expression of the oestrogen receptor (ER), progesterone receptor (PR), and HER2 has been reported in different areas of the same tumour. Furthermore, discordance, in terms of ER, PR and HER2 expression, has also been reported between primary tumours and their matched metastatic lesions. High-throughput molecular profiling studies have confirmed that spatial and temporal intratumour heterogeneity of breast cancers exist at a level beyond common expectations. We describe the different levels of tumour heterogeneity, and discuss the strategies that can be adopted by clinicians to tackle treatment response and resistance issues associated with such heterogeneity, including a rationally selected combination of agents that target driver mutations, the targeting of deleterious passenger mutations, identifying and eradicating the 'lethal' clone, targeting the tumour microenvironment, or using adaptive treatments and immunotherapy. The identification of the most-appropriate strategies and their implementation in the clinic will prove highly challenging and necessitate the adoption of radically new practices for the optimal clinical management of breast malignancies.

Analysis of Wnt signalling dynamics during colon crypt development in 3D culture

Many systems biology studies lack context-relevant data and as a consequence the predictive capabilities can be limited in developing targeted cancer therapeutics. Production of colon crypt in vitro is ideal for studying colon systems biology. This report presents the first production of, to our knowledge, physiologically-shaped, functional colon crypts in vitro (i.e. single crypts with cells expressing Mucin 2 and Chromogranin A). Time-lapsed monitoring of crypt formation revealed an increased frequency of single-crypt formation in the absence of noggin. Using quantitative 3D immunofluorescence of β-catenin and E-cadherin, spatial-temporal dynamics of these proteins in normal colon crypt cells stimulated with Wnt3A or inhibited by cycloheximide has been measured. Colon adenoma cultures established from APC^min/+ mouse have developmental differences and β-catenin spatial localization compared to normal crypts. Quantitative data describing the effects of signalling pathways and proteins dynamics for both normal and adenomatous colon crypts is now within reach to inform a systems approach to colon crypt biology.