Stochastic dynamics of metastasis formation

Chronic arsenic exposure induces malignant transformation of human HaCaT cells through both deterministic and stochastic changes in transcriptome expression

Biological processes are inherently stochastic, i.e., are partially driven by hard to predict random probabilistic processes. Carcinogenesis is driven both by stochastic and deterministic (predictable non-random) changes. However, very few studies systematically examine the contribution of stochastic events leading to cancer development. In differential gene expression studies, the established data analysis paradigms incentivize expression changes that are uniformly different across the experimental versus control groups, introducing preferential inclusion of deterministic changes at the expense of stochastic processes that might also play a crucial role in the process of carcinogenesis. In this study, we applied simple computational techniques to quantify: (i) The impact of chronic arsenic (iAs) exposure as well as passaging time on stochastic gene expression and (ii) Which genes were expressed deterministically and which were expressed stochastically at each of the three stages of cancer development. Using biological coefficient of variation as an empirical measure of stochasticity we demonstrate that chronic iAs exposure consistently suppressed passaging related stochastic gene expression at multiple time points tested, selecting for a homogenous cell population that undergo transformation. Employing multiple balanced removal of outlier data, we show that chronic iAs exposure induced deterministic and stochastic changes in the expression of unique set of genes, that populate largely unique biological pathways. Together, our data unequivocally demonstrate that both deterministic and stochastic changes in transcriptome-wide expression are critical in driving biological processes, pathways and networks towards clonal selection, carcinogenesis, and tumor heterogeneity.

Harnessing Normal and Engineered Mesenchymal Stem Cells Derived Exosomes for Cancer Therapy: Opportunity and Challenges

There remains a vital necessity for new therapeutic approaches to combat metastatic cancers, which cause globally over 8 million deaths per year. Mesenchymal stem cells (MSCs) display aptitude as new therapeutic choices for cancer treatment. Exosomes, the most important mediator of MSCs, regulate tumor progression. The potential of harnessing exosomes from MSCs (MSCs-Exo) in cancer therapy is now being documented. MSCs-Exo can promote tumor progression by affecting tumor growth, metastasis, immunity, angiogenesis, and drug resistance. However, contradictory evidence has suggested that MSCs-Exo suppress tumors through several mechanisms. Therefore, the exact association between MSCs-Exo and tumors remains controversial. Accordingly, the applications of MSCs-Exo as novel drug delivery systems and standalone therapeutics are being extensively explored. In addition, engineering MSCs-Exo for targeting tumor cells has opened a new avenue for improving the efficiency of antitumor therapy. However, effective implementation in the clinical trials will need the establishment of standards for MSCs-Exo isolation and characterization as well as loading and engineering methods. The studies outlined in this review highlight the pivotal roles of MSCs-Exo in tumor progression and the promising potential of MSCs-Exo as therapeutic drug delivery vehicles for cancer treatment.

Tumor-derived extracellular vesicles: The metastatic organotropism drivers

The continuous growing, spreading, and metastasis of tumor cells depend on intercellular communication within cells resident in a tissue environment. Such communication is mediated through the secretion of particles from tumor cells and resident cells known as extracellular vesicles (EVs) within a microenvironment. EVs are a heterogeneous population of membranous vesicles released from tumor cells that transfer many types of active biomolecules to recipient cells and induce physiologic and phenotypic alterations in the tissue environment. Spreading the 'seeds' of metastasis needs the EVs that qualify the 'soil' at distant sites to promote the progress of arriving tumor cells. Growing evidence indicates that EVs have vital roles in tumorigenesis, including pre-metastatic niche formation and organotropic metastasis. These EVs mediate organotropic metastasis by modifying the pre-metastatic microenvironment through different pathways including induction of phenotypic alternation and differentiation of cells, enrolment of distinct supportive stromal cells, up-regulation of the expression of pro-inflammatory genes, and induction of immunosuppressive status. However, instead of pre-metastatic niche formation, evidence suggests that EVs may mediate reawakening of dormant niches. Findings regarding EVs function in tumor metastasis have led to growing interests in the interdisciplinary significance of EVs, including targeted therapy, cell-free therapy, drug-delivery system, and diagnostic biomarker. In this review, we discuss EVs-mediated pre-metastatic niche formation and organotropic metastasis in visceral such as lung, liver, brain, lymph node, and bone with a focus on associated signaling, causing visceral environment hospitable for metastatic cells. Furthermore, we present an overview of the possible therapeutic application of EVs in cancer management.

Computational tools for assessing gene therapy under branching process models of mutation

Multitype branching processes are ideal for studying the population dynamics of stem cell populations undergoing mutation accumulation over the years following transplant. In such stochastic models, several quantities are of clinical interest as insertional mutagenesis carries the potential threat of leukemogenesis following gene therapy with autologous stem cell transplantation. In this paper, we develop a three-type branching process model describing accumulations of mutations in a population of stem cells distinguished by their ability for long-term self-renewal. Our outcome of interest is the appearance of a double-mutant cell, which carries a high potential for leukemic transformation. In our model, a single-hit mutation carries a slight proliferative advantage over a wild-type stem cells. We compute marginalized transition probabilities that allow us to capture important quantitative aspects of our model, including the probability of observing a double-hit mutant and relevant moments of a single-hit mutation population over time. We thoroughly explore the model behavior numerically, varying birth rates across the initial sizes and populations of wild type stem cells and single-hit mutants, and compare the probability of observing a double-hit mutant under these conditions. We find that increasing the number of single-mutants over wild-type particles initially present has a large effect on the occurrence of a double-mutant, and that it is relatively safe for single-mutants to be quite proliferative, provided the lentiviral gene addition avoids creating single mutants in the original insertion process. Our approach is broadly applicable to an important set of questions in cancer modeling and other population processes involving multiple stages, compartments, or types.

A mathematical model of the metastatic bottleneck predicts patient outcome and response to cancer treatment

Metastases are the main reason for cancer-related deaths. Initiation of metastases, where newly seeded tumor cells expand into colonies, presents a tremendous bottleneck to metastasis formation. Despite its importance, a quantitative description of metastasis initiation and its clinical implications is lacking. Here, we set theoretical grounds for the metastatic bottleneck with a simple stochastic model. The model assumes that the proliferation-to-death rate ratio for the initiating metastatic cells increases when they are surrounded by more of their kind. For a total of 159,191 patients across 13 cancer types, we found that a single cell has an extremely low median probability of successful seeding of the order of 10-8. With increasing colony size, a sharp transition from very unlikely to very likely successful metastasis initiation occurs. The median metastatic bottleneck, defined as the critical colony size that marks this transition, was between 10 and 21 cells. We derived the probability of metastasis occurrence and patient outcome based on primary tumor size at diagnosis and tumor type. The model predicts that the efficacy of patient treatment depends on the primary tumor size but even more so on the severity of the metastatic bottleneck, which is estimated to largely vary between patients. We find that medical interventions aiming at tightening the bottleneck, such as immunotherapy, can be much more efficient than therapies that decrease overall tumor burden, such as chemotherapy.

Stochastic Evolution of Pancreatic Cancer Metastases During Logistic Clonal Expansion

Despite recent progress in diagnostic and multimodal treatment approaches, most cancer deaths are still caused by metastatic spread and the subsequent growth of tumor cells in sites distant from the primary organ. So far, few quantitative studies are available that allow for the estimation of metastatic parameters and the evaluation of alternative treatment strategies. Most computational studies have focused on situations in which the tumor cell population expands exponentially over time; however, tumors may eventually be subject to resource and space limitations so that their growth patterns deviate from exponential growth to adhere to density-dependent growth models. In this study, we developed a stochastic evolutionary model of cancer progression that considers alterations in metastasis-related genes and intercellular growth competition leading to density effects described by logistic growth. Using this stochastic model, we derived analytical approximations for the time between the initiation of tumorigenesis and diagnosis, the expected number of metastatic sites, the total number of metastatic cells, the size of the primary tumor, and survival. Furthermore, we investigated the effects of drug administration and surgical resection on these quantities and predicted outcomes for different treatment regimens. Parameter values used in the analysis were estimated from data obtained from a pancreatic cancer rapid autopsy program. Our theoretical approach allows for flexible modeling of metastatic progression dynamics.

A mathematical model for the immune-mediated theory of metastasis

Accumulating experimental and clinical evidence suggest that the immune response to cancer is not exclusively anti-tumor. Indeed, the pro-tumor roles of the immune system  -  as suppliers of growth and pro-angiogenic factors or defenses against cytotoxic immune attacks, for example  -  have been long appreciated, but relatively few theoretical works have considered their effects. Inspired by the recently proposed "immune-mediated" theory of metastasis, we develop a mathematical model for tumor-immune interactions at two anatomically distant sites, which includes both anti- and pro-tumor immune effects, and the experimentally observed tumor-induced phenotypic plasticity of immune cells (tumor "education" of the immune cells). Upon confrontation of our model to experimental data, we use it to evaluate the implications of the immune-mediated theory of metastasis. We find that tumor education of immune cells may explain the relatively poor performance of immunotherapies, and that many metastatic phenomena, including metastatic blow-up, dormancy, and metastasis to sites of injury, can be explained by the immune-mediated theory of metastasis. Our results suggest that further work is warranted to fully elucidate the pro-tumor effects of the immune system in metastatic cancer.

Cancer recurrence times from a branching process model

As cancer advances, cells often spread from the primary tumor to other parts of the body and form metastases. This is the main cause of cancer related mortality. Here we investigate a conceptually simple model of metastasis formation where metastatic lesions are initiated at a rate which depends on the size of the primary tumor. The evolution of each metastasis is described as an independent branching process. We assume that the primary tumor is resected at a given size and study the earliest time at which any metastasis reaches a minimal detectable size. The parameters of our model are estimated independently for breast, colorectal, headneck, lung and prostate cancers. We use these estimates to compare predictions from our model with values reported in clinical literature. For some cancer types, we find a remarkably wide range of resection sizes such that metastases are very likely to be present, but none of them are detectable. Our model predicts that only very early resections can prevent recurrence, and that small delays in the time of surgery can significantly increase the recurrence probability.

Association of Uveal Melanoma Metastatic Rate With Stochastic Mutation Rate and Type of Mutation

Importance

It is necessary to understand the mechanisms of metastasis of uveal melanoma to advise patients and develop treatments for this tumor.

Objective

To examine the stochastic properties of primary uveal melanoma including the mutation rate as a function of tumor size and metastatic rate relative to the type of mutation.

Design, Setting, and Participants

We computed the mutation rate in different sized uveal melanomas using previously published large data sets. Tumor volume was estimated using the spherical cap method. We also calculated the metastatic rate using an updated data set of patients with uveal melanoma with known mutations in BAP1, SF3B1, and EIF1AX provided by the Rotterdam Ocular Melanoma Study Group. Data were analyzed from 2 studies, one taking place from August 25, 1970, to August 27, 2008, and the other taking place between 1993 and 2013. Data were analyzed between 2016 and 2017.

Main Outcomes and Measures

Mutation rates and metastic rates.

Results

Based on the 5-year metastatic rates, mutation rates ranged from 1.09 × 10-8 to 7.86 × 10-7 per cell division, using our calculation algorithm. A higher mutation rate was found for tumors with smaller thicknesses. EIF1AX mutations were not exclusive of other mutations because 2 cases with EIF1AX mutations and metastasis also had BAP1 mutations. None of the tumors with only an EIF1AX mutation metastasized. After plotting the yearly metastatic rate vs time after treatment, we observed a small peak at 1 year and a large peak at 3.5 years after treatment for BAP1 mutations, with peaks between 2 and 3 years and at 7 years for SF3B1 mutations.

Conclusions and Relevance

We observed a higher mutation rate for smaller tumors, which may be explained by a greater number of cell divisions occurring during the expansion phase of smaller uveal melanomas. Regarding time to clinically detected metastases, the first 2 peaks appear to be associated with BAP1-mutated tumors and the late peak to SF3B1-mutated tumors.

Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer

Brain metastases (BMs) are associated with poor prognosis in non-small cell lung cancer (NSCLC), but are only visible when large enough. Therapeutic decisions such as whole brain radiation therapy would benefit from patient-specific predictions of radiologically undetectable BMs. Here, we propose a mathematical modeling approach and use it to analyze clinical data of BM from NSCLC. Primary tumor growth was best described by a gompertzian model for the pre-diagnosis history, followed by a tumor growth inhibition model during treatment. Growth parameters were estimated only from the size at diagnosis and histology, but predicted plausible individual estimates of the tumor age (2.1-5.3 years). Multiple metastatic models were further assessed from fitting either literature data of BM probability (n = 183 patients) or longitudinal measurements of visible BMs in two patients. Among the tested models, the one featuring dormancy was best able to describe the data. It predicted latency phases of 4.4-5.7 months and onset of BMs 14-19 months before diagnosis. This quantitative model paves the way for a computational tool of potential help during therapeutic management.

Consecutive seeding and transfer of genetic diversity in metastasis

During metastasis, only a fraction of genetic diversity in a primary tumor is passed on to metastases. We calculate this fraction of transferred diversity as a function of the seeding rate between tumors. At one extreme, if a metastasis is seeded by a single cell, then it inherits only the somatic mutations present in the founding cell, so that none of the diversity in the primary tumor is transmitted to the metastasis. In contrast, if a metastasis is seeded by multiple cells, then some genetic diversity in the primary tumor can be transmitted. We study a multitype branching process of metastasis growth that originates from a single cell but over time receives additional cells. We derive a surprisingly simple formula that relates the expected diversity of a metastasis to the diversity in the pool of seeding cells. We calculate the probability that a metastasis is polyclonal. We apply our framework to published datasets for which polyclonality has been previously reported, analyzing 68 ovarian cancer samples, 31 breast cancer samples, and 8 colorectal cancer samples from 15 patients. For these clonally diverse metastases, under typical metastasis growth conditions, we find that 10 to 150 cells seeded each metastasis and left surviving lineages between initial formation and clinical detection.

A Mathematical Framework for Modelling the Metastatic Spread of Cancer

Cancer is a complex disease that starts with mutations of key genes in one cell or a small group of cells at a primary site in the body. If these cancer cells continue to grow successfully and, at some later stage, invade the surrounding tissue and acquire a vascular network, they can spread to distant secondary sites in the body. This process, known as metastatic spread, is responsible for around 90% of deaths from cancer and is one of the so-called hallmarks of cancer. To shed light on the metastatic process, we present a mathematical modelling framework that captures for the first time the interconnected processes of invasion and metastatic spread of individual cancer cells in a spatially explicit manner-a multigrid, hybrid, individual-based approach. This framework accounts for the spatiotemporal evolution of mesenchymal- and epithelial-like cancer cells, membrane-type-1 matrix metalloproteinase (MT1-MMP) and the diffusible matrix metalloproteinase-2 (MMP-2), and for their interactions with the extracellular matrix. Using computational simulations, we demonstrate that our model captures all the key steps of the invasion-metastasis cascade, i.e. invasion by both heterogeneous cancer cell clusters and by single mesenchymal-like cancer cells; intravasation of these clusters and single cells both via active mechanisms mediated by matrix-degrading enzymes (MDEs) and via passive shedding; circulation of cancer cell clusters and single cancer cells in the vasculature with the associated risk of cell death and disaggregation of clusters; extravasation of clusters and single cells; and metastatic growth at distant secondary sites in the body. By faithfully reproducing experimental results, our simulations support the evidence-based hypothesis that the membrane-bound MT1-MMP is the main driver of invasive spread rather than diffusible MDEs such as MMP-2.

Immune interconnectivity of anatomically distant tumors as a potential mediator of systemic responses to local therapy

Complex interactions occur between tumor and host immune system at each site in the metastatic setting, the outcome of which can determine behavior ranging from dormancy to rapid growth. An additional layer of complexity arises from the understanding that cytotoxic T cells can traffic through the host circulatory system. Coupling mathematical models of local tumor-immune dynamics and systemic T cell trafficking allows us to simulate the evolution of tumor and immune cell populations in anatomically distant sites following local therapy and thus computationally evaluate immune interconnectivity. Results suggest that the presence of a secondary site may either inhibit or promote growth of the primary, depending on the capacity for immune recruitment of each tumor and the resulting systemic redistribution of T cells. Treatment such as surgical resection and radiotherapy can be simulated to estimate both the decrease in tumor volume at the local treatment-targeted site, and the change in overall tumor burden and tumor growth trajectories across all sites. Qualitatively similar responses of distant tumors to local therapy (positive and negative abscopal effects) to those reported in the clinical setting were observed. Such findings may facilitate an improved understanding of general disease kinetics in the metastatic setting: if metastatic sites are interconnected through the immune system, truly local therapy does not exist.

Stochastic and Deterministic Models for the Metastatic Emission Process: Formalisms and Crosslinks

Although the detection of metastases radically changes prognosis of and treatment decisions for a cancer patient, clinically undetectable micrometastases hamper a consistent classification into localized or metastatic disease. This chapter discusses mathematical modeling efforts that could help to estimate the metastatic risk in such a situation. We focus on two approaches: (1) a stochastic framework describing metastatic emission events at random times, formalized via Poisson processes, and (2) a deterministic framework describing the micrometastatic state through a size-structured density function in a partial differential equation model. Three aspects are addressed in this chapter. First, a motivation for the Poisson process framework is presented and modeling hypotheses and mechanisms are introduced. Second, we extend the Poisson model to account for secondary metastatic emission. Third, we highlight an inherent crosslink between the stochastic and deterministic frameworks and discuss its implications. For increased accessibility the chapter is split into an informal presentation of the results using a minimum of mathematical formalism and a rigorous mathematical treatment for more theoretically interested readers.

Environmental factors in breast cancer invasion: a mathematical modelling review

This review presents a brief overview of breast cancer, focussing on its heterogeneity and the role of mathematical modelling and simulation in teasing apart the underlying biophysical processes. Following a brief overview of the main known pathophysiological features of ductal carcinoma, attention is paid to differential equation-based models (both deterministic and stochastic), agent-based modelling, multi-scale modelling, lattice-based models and image-driven modelling. A number of vignettes are presented where these modelling approaches have elucidated novel aspects of breast cancer dynamics, and we conclude by offering some perspectives on the role mathematical modelling can play in understanding breast cancer development, invasion and treatment therapies.

An exactly solvable, spatial model of mutation accumulation in cancer

One of the hallmarks of cancer is the accumulation of driver mutations which increase the net reproductive rate of cancer cells and allow them to spread. This process has been studied in mathematical models of well mixed populations, and in computer simulations of three-dimensional spatial models. But the computational complexity of these more realistic, spatial models makes it difficult to simulate realistically large and clinically detectable solid tumours. Here we describe an exactly solvable mathematical model of a tumour featuring replication, mutation and local migration of cancer cells. The model predicts a quasi-exponential growth of large tumours, even if different fragments of the tumour grow sub-exponentially due to nutrient and space limitations. The model reproduces clinically observed tumour growth times using biologically plausible rates for cell birth, death, and migration rates. We also show that the expected number of accumulated driver mutations increases exponentially in time if the average fitness gain per driver is constant, and that it reaches a plateau if the gains decrease over time. We discuss the realism of the underlying assumptions and possible extensions of the model.

The evolution of tumor metastasis during clonal expansion with alterations in metastasis driver genes

Metastasis is a leading cause of cancer-related deaths. Carcinoma generally initiates at a specific organ as a primary tumor, but eventually metastasizes and forms tumor sites in other organs. In this report, we developed a mathematical model of cancer progression with alterations in metastasis-related genes. In cases in which tumor cells acquire metastatic ability through two steps of genetic alterations, we derive formulas for the probability, the expected number, and the distribution of the number of metastases. Moreover, we investigate practical pancreatic cancer disease progression in cases in which both one and two steps of genetic alterations are responsible for metastatic formation. Importantly, we derive a mathematical formula for the survival outcome validated using clinical data as well as direct simulations. Our model provides theoretical insights into how invisible metastases distribute upon diagnosis with respect to growth rates, (epi)genetic alteration rates, metastatic rate, and detection size. Prediction of survival outcome using the formula is of clinical importance in terms of determining therapeutic strategies.

Modeling Spontaneous Metastasis following Surgery: An In Vivo-In Silico Approach

Rapid improvements in the detection and tracking of early-stage tumor progression aim to guide decisions regarding cancer treatments as well as predict metastatic recurrence in patients following surgery. Mathematical models may have the potential to further assist in estimating metastatic risk, particularly when paired with in vivo tumor data that faithfully represent all stages of disease progression. Herein, we describe mathematical analysis that uses data from mouse models of spontaneous metastasis developing after surgical removal of orthotopically implanted primary tumors. Both presurgical (primary tumor) growth and postsurgical (metastatic) growth were quantified using bioluminescence and were then used to generate a mathematical formalism based on general laws of the disease (i.e., dissemination and growth). The model was able to fit and predict pre/postsurgical data at the level of the individual as well as the population. Our approach also enabled retrospective analysis of clinical data describing the probability of metastatic relapse as a function of primary tumor size. In these data-based models, interindividual variability was quantified by a key parameter of intrinsic metastatic potential. Critically, our analysis identified a highly nonlinear relationship between primary tumor size and postsurgical survival, suggesting possible threshold limits for the utility of tumor size as a predictor of metastatic recurrence. These findings represent a novel use of clinically relevant models to assess the impact of surgery on metastatic potential and may guide optimal timing of treatments in neoadjuvant (presurgical) and adjuvant (postsurgical) settings to maximize patient benefit.

Quantifying metastatic inefficiency: rare genotypes versus rare dynamics

We introduce and solve a 'null model' of stochastic metastatic colonization. The model is described by a single parameter θ: the ratio of the rate of cell division to the rate of cell death for a disseminated tumour cell in a given secondary tissue environment. We are primarily interested in the case in which colonizing cells are poorly adapted for proliferation in the local tissue environment, so that cell death is more likely than cell division, i.e. θ < 1. We quantify the rare event statistics for the successful establishment of a metastatic colony of size N. For N >> 1, we find that the probability of establishment is exponentially rare, as expected, and yet the mean time for such rare events is of the form ~log (N)/(1 - θ) while the standard deviation of colonization times is ~1/(1 - θ). Thus, counter to naive expectation, for θ < 1, the average time for establishment of successful metastatic colonies decreases with decreasing cell fitness, and colonies seeded from lower fitness cells show less stochastic variation in their growth. These results indicate that metastatic growth from poorly adapted cells is rare, exponentially explosive and essentially deterministic. These statements are brought into sharper focus by the finding that the temporal statistics of the early stages of metastatic colonization from low-fitness cells (θ < 1) are statistically indistinguishable from those initiated from high-fitness cells (θ > 1), i.e. the statistics show a duality mapping (1 - θ) --> (θ - 1). We conclude our analysis with a study of heterogeneity in the fitness of colonising cells, and describe a phase diagram delineating parameter regions in which metastatic colonization is dominated either by low or high fitness cells, showing that both are plausible given our current knowledge of physiological conditions in human cancer.

Investigation of the essential role of platelet-tumor cell interactions in metastasis progression using an agent-based model

BACKGROUND

Metastatic tumors are a major source of morbidity and mortality for most cancers. Interaction of circulating tumor cells with endothelium, platelets and neutrophils play an important role in the early stages of metastasis formation. These complex dynamics have proven difficult to study in experimental models. Prior computational models of metastases have focused on tumor cell growth in a host environment, or prediction of metastasis formation from clinical data. We used agent-based modeling (ABM) to dynamically represent hypotheses of essential steps involved in circulating tumor cell adhesion and interaction with other circulating cells, examine their functional constraints, and predict effects of inhibiting specific mechanisms.

METHODS

We developed an ABM of Early Metastasis (ABMEM), a descriptive semi-mechanistic model that replicates experimentally observed behaviors of populations of circulating tumor cells, neutrophils, platelets and endothelial cells while incorporating representations of known surface receptor, autocrine and paracrine interactions. Essential downstream cellular processes were incorporated to simulate activation in response to stimuli, and calibrated with experimental data. The ABMEM was used to identify potential points of interdiction through examination of dynamic outcomes such as rate of tumor cell binding after inhibition of specific platelet or tumor receptors.

RESULTS

The ABMEM reproduced experimental data concerning neutrophil rolling over endothelial cells, inflammation-induced binding between neutrophils and platelets, and tumor cell interactions with these cells. Simulated platelet inhibition with anti-platelet drugs produced unstable aggregates with frequent detachment and re-binding. The ABMEM replicates findings from experimental models of circulating tumor cell adhesion, and suggests platelets play a critical role in this pre-requisite for metastasis formation. Similar effects were observed with inhibition of tumor integrin αV/β3. These findings suggest that anti-platelet or anti-integrin therapies may decrease metastasis by preventing stable circulating tumor cell adhesion.

CONCLUSION

Circulating tumor cell adhesion is a complex, dynamic process involving multiple cell-cell interactions. The ABMEM successfully captures the essential interactions necessary for this process, and allows for in-silico iterative characterization and invalidation of proposed hypotheses regarding this process in conjunction with in-vitro and in-vivo models. Our results suggest that anti-platelet therapies and anti-integrin therapies may play a promising role in inhibiting metastasis formation.

Modeling boundary conditions for balanced proliferation in metastatic latency

PURPOSE

Nearly half of cancer metastases become clinically evident five or more years after primary tumor treatment; thus, metastatic cells survived without emerging for extended periods. This dormancy has been explained by at least two countervailing scenarios: cellular quiescence and balanced proliferation; these entail dichotomous mechanistic etiologies. To examine the boundary parameters for balanced proliferation, we conducted in silico modeling.

EXPERIMENTAL DESIGN

To illuminate the balanced proliferation hypothesis, we explored the specific boundary probabilities under which proliferating micrometastases would remain dormant. A two-state Markov chain Monte Carlo model simulated micrometastatic proliferation and death according to stochastic survival probabilities. We varied these probabilities across 100 simulated patients each with 1,000 metastatic deposits and documented whether the micrometastases exceeded one million cells, died out, or remained dormant (survived 1,218 generations).

RESULTS

The simulations revealed a narrow survival probability window (49.7-50.8%) that allowed for dormancy across a range of starting cell numbers, and even then for only a small fraction of micrometastases. The majority of micrometastases died out quickly even at survival probabilities that led to rapid emergence of a subset of micrometastases. Within dormant metastases, cell populations depended sensitively on small survival probability increments.

CONCLUSIONS

Metastatic dormancy as explained solely by balanced proliferation is bounded by very tight survival probabilities. Considering the far larger survival variability thought to attend fluxing microenvironments, it is more probable that these micrometastatic nodules undergo at least periods of quiescence rather than exclusively being controlled by balanced proliferation.

Applying ecological and evolutionary theory to cancer: a long and winding road

Since the mid 1970s, cancer has been described as a process of Darwinian evolution, with somatic cellular selection and evolution being the fundamental processes leading to malignancy and its many manifestations (neoangiogenesis, evasion of the immune system, metastasis, and resistance to therapies). Historically, little attention has been placed on applications of evolutionary biology to understanding and controlling neoplastic progression and to prevent therapeutic failures. This is now beginning to change, and there is a growing international interest in the interface between cancer and evolutionary biology. The objective of this introduction is first to describe the basic ideas and concepts linking evolutionary biology to cancer. We then present four major fronts where the evolutionary perspective is most developed, namely laboratory and clinical models, mathematical models, databases, and techniques and assays. Finally, we discuss several of the most promising challenges and future prospects in this interdisciplinary research direction in the war against cancer.

Quantifying introgression risk with realistic population genetics

Introgression is the permanent incorporation of genes from the genome of one population into another. This can have severe consequences, such as extinction of endemic species, or the spread of transgenes. Quantification of the risk of introgression is an important component of genetically modified crop regulation. Most theoretical introgression studies aimed at such quantification disregard one or more of the most important factors concerning introgression: realistic genetical mechanisms, repeated invasions and stochasticity. In addition, the use of linkage as a risk mitigation strategy has not been studied properly yet with genetic introgression models. Current genetic introgression studies fail to take repeated invasions and demographic stochasticity into account properly, and use incorrect measures of introgression risk that can be manipulated by arbitrary choices. In this study, we present proper methods for risk quantification that overcome these difficulties. We generalize a probabilistic risk measure, the so-called hazard rate of introgression, for application to introgression models with complex genetics and small natural population sizes. We illustrate the method by studying the effects of linkage and recombination on transgene introgression risk at different population sizes.

Modeling the connection between primary and metastatic tumors

We put forward a model for cancer metastasis as a migration phenomenon between tumor cell populations coexisting and evolving in two different habitats. One of them is a primary tumor and the other one is a secondary or metastatic tumor. The evolution of the different cell phenotype populations in each habitat is described by means of a simple quasispecies model allowing for a cascade of mutations between the different phenotypes in each habitat. The cell migration event is supposed to be unidirectional and take place continuously in time. The possible clinical outcomes of the model depending on the parameter space are analyzed and the effect of the resection of the primary tumor is studied.

Computational modeling of pancreatic cancer reveals kinetics of metastasis suggesting optimum treatment strategies

Pancreatic cancer is a leading cause of cancer-related death, largely due to metastatic dissemination. We investigated pancreatic cancer progression by utilizing a mathematical framework of metastasis formation together with comprehensive data of 228 patients, 101 of whom had autopsies. We found that pancreatic cancer growth is initially exponential. After estimating the rates of pancreatic cancer growth and dissemination, we determined that patients likely harbor metastases at diagnosis and predicted the number and size distribution of metastases as well as patient survival. These findings were validated in an independent database. Finally, we analyzed the effects of different treatment modalities, finding that therapies that efficiently reduce the growth rate of cells earlier in the course of treatment appear to be superior to upfront tumor resection. These predictions can be validated in the clinic. Our interdisciplinary approach provides insights into the dynamics of pancreatic cancer metastasis and identifies optimum therapeutic interventions.

Computational modeling of pancreatic cancer reveals kinetics of metastasis suggesting optimum treatment strategies

Pancreatic cancer is a leading cause of cancer-related death, largely due to metastatic dissemination. We investigated pancreatic cancer progression by utilizing a mathematical framework of metastasis formation together with comprehensive data of 228 patients, 101 of whom had autopsies. We found that pancreatic cancer growth is initially exponential. After estimating the rates of pancreatic cancer growth and dissemination, we determined that patients likely harbor metastases at diagnosis and predicted the number and size distribution of metastases as well as patient survival. These findings were validated in an independent database. Finally, we analyzed the effects of different treatment modalities, finding that therapies that efficiently reduce the growth rate of cells earlier in the course of treatment appear to be superior to upfront tumor resection. These predictions can be validated in the clinic. Our interdisciplinary approach provides insights into the dynamics of pancreatic cancer metastasis and identifies optimum therapeutic interventions.

Quantifying time-inhomogeneous stochastic introgression processes with hazard rates

Introgression is the permanent incorporation of genes from one population into another through hybridization and backcrossing. It is currently of particular concern as a possible mechanism for the spread of modified crop genes to wild populations. The hazard rate is the probability per time unit that such an escape takes place, given that it has not happened before. It is a quantitative measure of introgression risk that takes the stochastic elements inherent in introgression processes into account. We present a methodology to calculate the hazard rate for situations with time-varying gene flow from a crop to a large recipient wild population. As an illustration, several types of time-inhomogeneity are examined, including deterministic periodicity as well as random variation. Furthermore, we examine the effects of an extended fitness bottleneck of hybrids and backcrosses in combination with time-varying gene flow. It is found that bottlenecks decrease the hazard rate, but also slow down and delay its changes in reaction to changes in gene flow. Furthermore, we find that random variation in gene flow generates a lower hazard rate than analogous deterministic variation. We discuss the implications of our findings for crop management and introgression risk assessment.

Solving the puzzle of metastasis: the evolution of cell migration in neoplasms

Background

Metastasis represents one of the most clinically important transitions in neoplastic progression. The evolution of metastasis is a puzzle because a metastatic clone is at a disadvantage in competition for space and resources with non-metastatic clones in the primary tumor. Metastatic clones waste some of their reproductive potential on emigrating cells with little chance of establishing metastases. We suggest that resource heterogeneity within primary tumors selects for cell migration, and that cell emigration is a by-product of that selection.

Methods and Findings

We developed an agent-based model to simulate the evolution of neoplastic cell migration. We simulated the essential dynamics of neoangiogenesis and blood vessel occlusion that lead to resource heterogeneity in neoplasms. We observed the probability and speed of cell migration that evolves with changes in parameters that control the degree of spatial and temporal resource heterogeneity. Across a broad range of realistic parameter values, increasing degrees of spatial and temporal heterogeneity select for the evolution of increased cell migration and emigration.

Conclusions

We showed that variability in resources within a neoplasm (e.g. oxygen and nutrients provided by angiogenesis) is sufficient to select for cells with high motility. These cells are also more likely to emigrate from the tumor, which is the first step in metastasis and the key to the puzzle of metastasis. Thus, we have identified a novel potential solution to the puzzle of metastasis.

Stochastic dynamics of cancer initiation

Most human cancer types result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Once the first change (or changes) have arisen, tumorigenesis is initiated and the subsequent emergence of additional alterations drives progression to more aggressive and ultimately invasive phenotypes. Elucidation of the dynamics of cancer initiation is of importance for an understanding of tumor evolution and cancer incidence data. In this paper, we develop a novel mathematical framework to study the processes of cancer initiation. Cells at risk of accumulating oncogenic mutations are organized into small compartments of cells and proliferate according to a stochastic process. During each cell division, an (epi)genetic alteration may arise which leads to a random fitness change, drawn from a probability distribution. Cancer is initiated when a cell gains a fitness sufficiently high to escape from the homeostatic mechanisms of the cell compartment. To investigate cancer initiation during a human lifetime, a 'race' between this fitness process and the aging process of the patient is considered; the latter is modeled as a second stochastic Markov process in an aging dimension. This model allows us to investigate the dynamics of cancer initiation and its dependence on the mutational fitness distribution. Our framework also provides a methodology to assess the effects of different life expectancy distributions on lifetime cancer incidence. We apply this methodology to colorectal tumorigenesis while considering life expectancy data of the US population to inform the dynamics of the aging process. We study how the probability of cancer initiation prior to death, the time until cancer initiation, and the mutational profile of the cancer-initiating cell depends on the shape of the mutational fitness distribution and life expectancy of the population.

The origins and implications of intratumor heterogeneity

Human tumors often display startling intratumor heterogeneity in various features including histology, gene expression, genotype, and metastatic and proliferative potential. This phenotypic and genetic heterogeneity plays an important role in neoplasia, cancer progression, and therapeutic resistance. In this issue of the journal (beginning on page 1388), Merlo et al. report their use of molecular data from 239 patients with Barrett's esophagus to evaluate the propensity of major diversity indices for predicting progression to esophageal adenocarcinoma. This work helps elucidate the implications of molecular heterogeneity for the evolution of neoplasia.

Evolutionary dynamics of tumor progression with random fitness values

Most human tumors result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Mutations that confer a fitness advantage to the cell are known as driver mutations and are causally related to tumorigenesis. Other mutations, however, do not change the phenotype of the cell or even decrease cellular fitness. While much experimental effort is being devoted to the identification of the functional effects of individual mutations, mathematical modeling of tumor progression generally considers constant fitness increments as mutations are accumulated. In this paper we study a mathematical model of tumor progression with random fitness increments. We analyze a multi-type branching process in which cells accumulate mutations whose fitness effects are chosen from a distribution. We determine the effect of the fitness distribution on the growth kinetics of the tumor. This work contributes to a quantitative understanding of the accumulation of mutations leading to cancer.

The evolution of tumor metastases during clonal expansion

Cancer is a leading cause of morbidity and mortality in many countries. Solid tumors generally initiate at one particular site called the primary tumor, but eventually disseminate and form new colonies in other organs. The development of such metastases greatly diminishes the potential for a cure of patients and is thought to represent the final stage of the multi-stage progression of human cancer. The concept of early metastatic dissemination, however, postulates that cancer cell spread might arise early during the development of a tumor. It is important to know whether metastases are present at diagnosis since this determines treatment strategies and outcome. In this paper, we design a stochastic mathematical model of the evolution of tumor metastases in an expanding cancer cell population. We calculate the probability of metastasis at a given time during tumor evolution, the expected number of metastatic sites, and the total number of cancer cells as well as metastasized cells. Furthermore, we investigate the effect of drug administration and tumor resection on these quantities and predict the survival time of cancer patients. The model presented in this paper allows us to determine the probability and number of metastases at diagnosis and to identify the optimum treatment strategy to maximally prolong survival of cancer patients.

Evolutionary theory of cancer

As Theodosius Dobzhansky famously noted in 1973, "Nothing in biology makes sense except in the light of evolution," and cancer is no exception to this rule. Our understanding of cancer initiation, progression, treatment, and resistance has advanced considerably by regarding cancer as the product of evolutionary processes. Here we review the literature of mathematical models of cancer evolution and provide a synthesis and discussion of the field.

Empirically-based estimates for the burden of subclinical metastases

PURPOSE

To describe the frequency distribution for the number of residual subclinical metastatic tumor cells after removal of the primary cancer.

MATERIALS AND METHODS

Previously obtained autopsy, surgical pathological and laboratory data were used to characterize the size and number distributions for hematogenous and lymphatic metastases. Monte Carlo simulations were used to estimate the numbers of residual tumor cells based upon the assumption of a lognormal distribution for the sizes of metastases and Poisson, Poisson negative binomial, or negative binomial distributed numbers of metastases (corresponding to lymphatic metastases within individuals, hematogenous metastases within individuals, and lymphatic metastases within populations, respectively).

RESULTS

In each of the scenarios the resultant distribution for the numbers of subclinical tumor cells was unimodal and positively skewed, with a tail extending to the higher numbers of metastases. When plotted with equal sized counting bins and according the logarithm of the number of tumor cells, the distributions showed deviations from the normal form no greater than several percentage points--a result considered acceptable given the variabilities inherent to metastasis data.

CONCLUSIONS

The distribution for the number of residual subclinical metastases may be extrapolated from data and models derived from the size and number distributions for metastases. In the absence of a closed form description for this distribution, the lognormal distribution could provide a crude, but practical, approximation for cases limited to occult microscopic residual disease. These analyses will facilitate the definition of the dose-response for the adjuvant therapy of subclinical metastases.

Nanovehicular intracellular delivery systems

This article provides an overview of principles and barriers relevant to intracellular drug and gene transport, accumulation and retention (collectively called as drug delivery) by means of nanovehicles (NV). The aim is to deliver a cargo to a particular intracellular site, if possible, to exert a local action. Some of the principles discussed in this article apply to noncolloidal drugs that are not permeable to the plasma membrane or to the blood-brain barrier. NV are defined as a wide range of nanosized particles leading to colloidal objects which are capable of entering cells and tissues and delivering a cargo intracelullarly. Different localization and targeting means are discussed. Limited discussion on pharmacokinetics and pharmacodynamics is also presented. NVs are contrasted to micro-delivery and current nanotechnologies which are already in commercial use. Newer developments in NV technologies are outlined and future applications are stressed. We also briefly review the existing modeling tools and approaches to quantitatively describe the behavior of targeted NV within the vascular and tumor compartments, an area of particular importance. While we list "elementary" phenomena related to different level of complexity of delivery to cancer, we also stress importance of multi-scale modeling and bottom-up systems biology approach.

Towards a multiscale model of colorectal cancer

Colorectal cancer (CRC) is one of the best characterised cancers, with extensive data documenting the sequential gene mutations that underlie its development. Complementary datasets are also being generated describing changes in protein and RNA expression, tumour biology and clinical outcome. Both the quantity and the variety of information are inexorably increasing and there is now an accompanying need to integrate these highly disparate datasets. In this article we aim to explain why we believe that mathematical modelling represents a natural tool or language with which to integrate these data and, in so doing, to provide insight into CRC.

Dynamics of metastasis suppressor gene inactivation

For most cancer cell types, the acquisition of metastatic ability leads to clinically incurable disease. Twelve metastasis suppressor genes (MSGs) have been identified that reduce the metastatic propensity of cancer cells. If these genes are inactivated in both alleles, metastatic ability is promoted. Here, we develop a mathematical model of the dynamics of MSG inactivation and calculate the expected number of metastases formed by a tumor. We analyse the effects of increased mutation rates and different fitness values of cells with one or two inactivated alleles on the ability of a tumor to form metastases. We find that mutations that are negatively selected in the main tumor are unlikely to be responsible for the majority of metastases produced by a tumor. Most metastases-causing mutations will be present in all (or most) cells in the main tumor.