Mathematical modeling of the metastatic process

Mathematical modeling in cancer has been growing in popularity and impact since its inception in 1932. The first theoretical mathematical modeling in cancer research was focused on understanding tumor growth laws and has grown to include the competition between healthy and normal tissue, carcinogenesis, therapy and metastasis. It is the latter topic, metastasis, on which we will focus this short review, specifically discussing various computational and mathematical models of different portions of the metastatic process, including: the emergence of the metastatic phenotype, the timing and size distribution of metastases, the factors that influence the dormancy of micrometastases and patterns of spread from a given primary tumor.Comment: 24 pages, 6 figures, Revie

The Therapeutic Implications of Plasticity of the Cancer Stem Cell Phenotype

The cancer stem cell hypothesis suggests that tumors contain a small population of cancer cells that have the ability to undergo symmetric self-renewing cell division. In tumors that follow this model, cancer stem cells produce various kinds of specified precursors that divide a limited number of times before terminally differentiating or undergoing apoptosis. As cells within the tumor mature, they become progressively more restricted in the cell types to which they can give rise. However, in some tumor types, the presence of certain extra- or intracellular signals can induce committed cancer progenitors to revert to a multipotential cancer stem cell state. In this paper, we design a novel mathematical model to investigate the dynamics of tumor progression in such situations, and study the implications of a reversible cancer stem cell phenotype for therapeutic interventions. We find that higher levels of dedifferentiation substantially reduce the effectiveness of therapy directed at cancer stem cells by leading to higher rates of resistance. We conclude that plasticity of the cancer stem cell phenotype is an important determinant of the prognosis of tumors. This model represents the first mathematical investigation of this tumor trait and contributes to a quantitative understanding of cancer

Evolution of Resistance to Targeted Anti-Cancer Therapies during Continuous and Pulsed Administration Strategies

The discovery of small molecules targeted to specific oncogenic pathways has revolutionized anti-cancer therapy. However, such therapy often fails due to the evolution of acquired resistance. One long-standing question in clinical cancer research is the identification of optimum therapeutic administration strategies so that the risk of resistance is minimized. In this paper, we investigate optimal drug dosing schedules to prevent, or at least delay, the emergence of resistance. We design and analyze a stochastic mathematical model describing the evolutionary dynamics of a tumor cell population during therapy. We consider drug resistance emerging due to a single (epi)genetic alteration and calculate the probability of resistance arising during specific dosing strategies. We then optimize treatment protocols such that the risk of resistance is minimal while considering drug toxicity and side effects as constraints. Our methodology can be used to identify optimum drug administration schedules to avoid resistance conferred by one (epi)genetic alteration for any cancer and treatment type

The Probable Cell of Origin of NF1- and PDGF-Driven Glioblastomas

Primary glioblastomas are subdivided into several molecular subtypes. There is an ongoing debate over the cell of origin for these tumor types where some suggest a progenitor while others argue for a stem cell origin. Even within the same molecular subgroup, and using lineage tracing in mouse models, different groups have reached different conclusions. We addressed this problem from a combined mathematical modeling and experimental standpoint. We designed a novel mathematical framework to identify the most likely cells of origin of two glioma subtypes. Our mathematical model of the unperturbed in vivo system predicts that if a genetic event contributing to tumor initiation imparts symmetric self-renewing cell division (such as PDGF overexpression), then the cell of origin is a transit amplifier. Otherwise, the initiating mutations arise in stem cells. The mathematical framework was validated with the RCAS/tv-a system of somatic gene transfer in mice. We demonstrated that PDGF-induced gliomas can be derived from GFAP-expressing cells of the subventricular zone or the cortex (reactive astrocytes), thus validating the predictions of our mathematical model. This interdisciplinary approach allowed us to determine the likelihood that individual cell types serve as the cells of origin of gliomas in an unperturbed system

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.Comment: 26 pages, 9 figures, 4 table

Rethinking therapeutic strategies in cancer: wars, fields, anomalies and monsters

This article argues that the excessive focus on cancer as an insidious living defect that needs to be destroyed has obscured the fact that cancer develops inside human beings. Therefore, in order to contribute to debates about new cancer therapies, we argue that it is important to gain a broader understanding of what cancer is and how it might be otherwise. First, in order to reframe the debate, we utilize Pierre Bourdieu’s field analysis in order to gain a stronger understanding of the structure of the (sub)field of cancer research. In doing so, we are able to see that those in a dominant position in the field, with high levels of scientific capital at their disposal, are in the strongest position to determine the type of research that is carried out and, more significantly, how cancer is perceived. Field analysis enables us to gain a greater understanding of the complex interplay between the field of science (and, more specifically, the subfield of cancer research) and broader sources of power. Second, we draw attention to new possible ways of understanding cancer in its evolutionary context. One of the problems facing cancer research is the narrow time frame within which cancer is perceived: the lives of cancer cells are considered from the moment the cells initially change. In contrast, the approach put forward here requires a different way of thinking: we take a longer view and consider cancer as a living entity, with cancer perceived as anomalous rather than abnormal. Third, we theorize the possibility of therapeutic strategies that might involve the redirection (rather than the eradication) of cancer cells. This approach also necessitates new ways of perceiving cancer

Stochastic Theory of Early Viral Infection: Continuous versus Burst Production of Virions

Viral production from infected cells can occur continuously or in a burst that generally kills the cell. For HIV infection, both modes of production have been suggested. Standard viral dynamic models formulated as sets of ordinary differential equations can not distinguish between these two modes of viral production, as the predicted dynamics is identical as long as infected cells produce the same total number of virions over their lifespan. Here we show that in stochastic models of viral infection the two modes of viral production yield different early term dynamics. Further, we analytically determine the probability that infections initiated with any number of virions and infected cells reach extinction, the state when both the population of virions and infected cells vanish, and show this too has different solutions for continuous and burst production. We also compute the distributions of times to establish infection as well as the distribution of times to extinction starting from both a single virion as well as from a single infected cell for both modes of virion production

Recasting the cancer stem cell hypothesis: Unification using a continuum model of microenvironmental forces

Purpose of review Here, we identify shortcomings of standard compartment-based mathematical models of cancer stem-cells, and propose a continuous formalism which includes the tumor microenvironment. Recent findings Stem-cell models of tumor growth have provided explanations for various phenomena in oncology including, metastasis, drug- and radio-resistance, and functional heterogeneity in the face of genetic homogeneity. While some of the newer models allow for plasticity, or de-differentiation, there is no consensus on the mechanisms driving this. Recent experimental evidence suggests that tumor microenvironment factors like hypoxia, acidosis, and nutrient deprivation have causative roles. Summary To settle the dissonance between the mounting experimental evidence surrounding the effects of the microenvironment on tumor stemness, we propose a continuous mathematical model where we model microenvironmental perturbations like forces, which then shape the distribution of stemness within the tumor. We propose methods by which to systematically measure and characterize these forces, and show results of a simple experiment which support our claims

Optimal reaction coordinate as a biomarker for the dynamics of recovery from kidney transplant.

The evolution of disease or the progress of recovery of a patient is a complex process, which depends on many factors. A quantitative description of this process in real-time by a single, clinically measurable parameter (biomarker) would be helpful for early, informed and targeted treatment. Organ transplantation is an eminent case in which the evolution of the post-operative clinical condition is highly dependent on the individual case. The quality of management and monitoring of patients after kidney transplant often determines the long-term outcome of the graft. Using NMR spectra of blood samples, taken at different time points from just before to a week after surgery, we have shown that a biomarker can be found that quantitatively monitors the evolution of a clinical condition. We demonstrate that this is possible if the dynamics of the process is considered explicitly: the biomarker is defined and determined as an optimal reaction coordinate that provides a quantitatively accurate description of the stochastic recovery dynamics. The method, originally developed for the analysis of protein folding dynamics, is rigorous, robust and general, i.e., it can be applied in principle to analyze any type of biological dynamics. Such predictive biomarkers will promote improvement of long-term graft survival after renal transplantation, and have potentially unlimited applications as diagnostic tools

Mathematical Modelling as a Proof of Concept for MPNs as a Human Inflammation Model for Cancer Development

<p><b>Left:</b> Typical development in stem cells (top panel A) and mature cells (bottom panel B). Healthy hematopoietic cells (full blue curves) dominate in the early phase where the number of malignant cells (stipulated red curves) are few. The total number of cells is also shown (dotted green curves). When a stem cell mutates without repairing mechanisms, a slowly increasing exponential growth starts. At a certain stage, the malignant cells become dominant, and the healthy hematopoietic cells begin to show a visible decline. Finally, the composition between the cell types results in a takeover by the malignant cells, leading to an exponential decline in hematopoietic cells and ultimately their extinction. The development is driven by an approximately exponential increase in the MPN stem cells, and the development is closely followed by the mature MPN cells. <b>Right:</b> B)The corresponding allele burden (7%, 33% and 67% corresponding to ET, PV, and PMF, respectively) defined as the ratio of MPN mature cells to the total number of mature cells.</p