A simple mathematical model based on the cancer stem cell hypothesis suggests kinetic commonalities in solid tumor growth

Eta polycaprolactone (ε-PCL) implants appear to cause a partial differentiation of breast cancer lung metastasis in a murine model

BACKGROUND

Cells in every epithelium can be roughly divided in three compartments: stem cell (SC) compartment, transient amplifying cell (TA) compartment and terminally differentiated (TD) compartment. Maturation of stem cells is characterized by epithelial stromal interaction and sequential maturational movement of stem cell's progeny through those compartments. In this work we hypothesize that providing an artificial stroma, which murine breast cancer metastatic cells can infiltrate, will induce their differentiation.

METHODS

BALB/c female mice were injected with 10⁶ isogenic 4T1 breast cancer cells labeled with GFP. After 20 days primary tumors were removed, and artificial ε-PCL implants were implanted on the contralateral side. After 10 more days mice were sacrificed and implants along with lung tissue were harvested. Mice were divided in four groups: tumor removal with sham implantation surgery (n = 5), tumor removal with ε-PCL implant (n = 5), tumor removal with VEGF enriched ε-PCL implant (n = 7) and mice without tumor with VEGF enriched ε-PCL implant (n = 3). Differentiational status of GFP + cells was assessed by Ki67 and activated caspase 3 expression, thus dividing the population in SC like cells (Ki67^+/dim aCasp3^-), TA like cells (Ki67^+/dim aCasp3^+/dim) and TD like cells (Ki67^- aCasp3^+/dim) on flow cytometry.

RESULTS

Lung metastatic load was reduced by 33% in mice with simple ε-PCL implant when compared to tumor bearing group with no implant. Mice with VEGF enriched implants had 108% increase in lung metastatic load in comparison to tumor bearing mice with no implants. Likewise, amount of GFP + cells was higher in simple ε-PCL implant in comparison to VEGF enriched implants. Differentiation-wise, process of metastasizing to lungs reduces the average fraction of SC like cells when compared to primary tumor. This effect is made more uniform by both kinds of ε-PCL implants. The opposite process is mirrored in TA like cells compartment when it comes to averages. Effects of both types of implants on TD like cells were negligible. Furthermore, if gene expression signatures that mimic tissue compartments are analyzed in human breast cancer metastases, it turns out that TA signature is associated with increased survival probability.

CONCLUSION

ε-PCL implants without VEGF can reduce metastatic loads in lungs, after primary tumor removal. Both types of implants cause lung metastasis differentiation by shifting cancer cells from SC to TA compartment, leaving the TD compartment unaffected.

Evolution of cancer stem cell lineage involving feedback regulation

Tumor emergence and progression is a complex phenomenon that assumes special molecular and cellular interactions. The hierarchical structuring and communication via feedback signaling of different cell types, which are categorized as the stem, progenitor, and differentiated cells in dependence of their maturity level, plays an important role. Under healthy conditions, these cells build a dynamical system that is responsible for facilitating the homeostatic regulation of the tissue. Generally, in this hierarchical setting, stem and progenitor cells are yet likely to undergo a mutation, when a cell divides into two daughter cells. This may lead to the development of abnormal characteristics, i.e. mutation in the cell, yielding an unrestrained number of cells. Therefore, the regulation of a stem cell’s proliferation and differentiation rate is crucial for maintaining the balance in the overall cell population. In this paper, a maturity based mathematical model with feedback regulation is formulated for healthy and mutated cell lineages. It is given in the form of coupled ordinary and partial differential equations. The focus is laid on the dynamical effects resulting from acquiring a mutation in the hierarchical structure of stem, progenitor and fully differentiated cells. Additionally, the effects of nonlinear feedback regulation from mature cells into both stem and progenitor cell populations have been inspected. The steady-state solutions of the model are derived analytically. Numerical simulations and results based on a finite volume scheme underpin various expected behavioral patterns of the homeostatic regulation and cancer evolution. For instance, it has been found that the mutated cells can experience significant growth even with a single somatic mutation, but under homeostatic regulation acquire a steady-state and thus, ensuing healthy cell population to either a steady-state or a lower cell concentration. Furthermore, the model behavior has been validated with different experimentally measured tumor values from the literature.

MATHEMATICAL CHARACTERIZATION OF HETEROGENEITY IN A CANCER STEM CELL DRIVEN TUMOR GROWTH MODEL WITH NONLINEAR SELF-RENEWAL

The detection, in a wide variety of cancer types, of a population of highly tumorigenic cells that exhibit self-renewal and multipotency, which are hallmarks of stem cells, has transformed the current view of tumor initiation, progression, and treatment. Here, we develop and analyze a mathematical model for tumor growth that is based on the current biological understanding of the processes that underlie cellular expansion under the hierarchical guidelines of the cancer stem cell (CSC) hypothesis. Important features of the model include (i) a nonlinear probability of CSC self-renewal that reflects the fact that this key type of stem cell division can be regulated by extrinsic and intrinsic chemical signaling as well as environmental (niche) constraints and (ii) an amplification factor that captures the transient amplifying divisions that are a defining characteristic of progenitor cells. We present a thorough mathematical analysis of the model and highlight the conditions required for tumors to evolve toward either bounded or exponential growth. Numerical simulations further illustrate the impact of the various parameters on the tumor growth rate and on the heterogeneous cellular composition, which varies during progression.

A Mathematical Model of Average Dynamics in a Stem Cell Hierarchy Suggests the Combinatorial Targeting of Cancer Stem Cells and Progenitor Cells as a Potential Strategy against Tumor Growth

Simple Summary

Cancer stem cell (CSC) directed therapies have been increasingly developed during the last years. However, some reported experiments using this strategy showed that although delayed tumor growth was observed, the tumor was not completely eliminated. Here, we hypothesize that the simultaneous targeting of CSCs and progenitor cells of intermediate phenotype may represent a better strategy against tumor growth. We aimed to fit a mathematical model, consistent with the CSC hypothesis, to reported experimental data resulting from CSC direct targeting. This is a minimal model of average tumor dynamics that could aid in the visualization of the overall tumor growth when different subpopulations of tumor cells are targeted. We show that combination therapy during a time lapse that ensures eradication of CSCs and progenitor cells in a stem cell hierarchy controlled tumor relapse. Testing this hypothesis in vivo may help to discriminate among other possibilities of tumor burden.

Abstract

The cancer stem cell hypothesis states that tumors are maintained by a small subpopulation of stem-like cells, often called cancer stem cells (CSCs) or tumor initiating cells. CSCs can self-renew and give rise to more differentiated cells, which comprise the bulk of the tumor. In addition, CSCs are resistant to conventional therapy, which suggests that they are responsible for tumor relapse. This has led researchers to increase efforts to develop directed therapies against CSCs. However, some experiments in mice have shown that the elimination of CSCs might not ensure tumor eradication. This may be due to different events, such as residual CSCs after treatment, the plasticity of cells within the tumor, the presence of different CSCs having their own hierarchy within the same tumor, and the ability of more differentiated cells to maintain the disease, among others. Trying to decipher this complexity may benefit from dissecting the whole in its parts. Here, we hypothesize that tumor relapse after the selective targeting of CSCs may be due to intermediate progenitor (P) cells that can maintain the tumor volume. In order to support the hypothesis, we implemented a mathematical model derived using pseudo-reactions representing the events of each cell subpopulation within the tumor. We aimed to test if a minimal unidirectional hierarchical model consisting of CSCs, P, and terminally differentiated (D) cells could be adjusted to experimental data for selective CSC targeting. We further evaluated therapies ranging from nonselective to specifically directed and combination therapy. We found that selective killing of the CSC compartment has a delaying effect on the overall exponential tumor growth, but was not able to eliminate the disease. We show that therapy that targets both CSCs and intermediate progenitor (P) cells with a sufficient capacity to proliferate and differentiate could represent a more efficient treatment option for tumor depletion. Testing this hypothesis in vivo may allow us to discriminate within the array of possibilities of tumor relapse, and further open the idea of combination therapy against different subpopulations of tumor cells instead of segregating CSCs and bulk tumor cells.

A Mathematical Model to Predict Diagnostic Periods for Secondary Distant Metastases in Patients with ER/PR/HER2/Ki-67 Subtypes of Breast Cancer

Previously, a consolidated mathematical model of primary tumor (PT) growth and secondary distant metastasis (sdMTS) growth in breast cancer (BC) (CoMPaS) was presented. The aim was to detect the diagnostic periods for visible sdMTS via CoMPaS in patients with different subtypes ER/PR/HER2/Ki-67 (Estrogen Receptor/Progesterone Receptor/Human Epidermal growth factor Receptor 2/Ki-67 marker) of breast cancer. CoMPaS is based on an exponential growth model and complementing formulas, and the model corresponds to the tumor-node-metastasis (TNM) staging system and BC subtypes (ER/PR/HER2/Ki-67). The CoMPaS model reflects (1) the subtypes of BC, such as ER/PR/HER2/Ki-67, and (2) the growth processes of the PT and sdMTSs in BC patients without or with lymph node metastases (MTSs) in accordance with the eighth edition American Joint Committee on Cancer prognostic staging system for breast cancer. CoMPaS correctly describes the growth of the PT in the ER/PR/HER2/Ki-67 subtypes of BC patients and helps to calculate the different diagnostic periods, depending on the tumor volume doubling time of sdMTS, when sdMTSs might appear. CoMPaS and the corresponding software tool can help (1) to start the early treatment of small sdMTSs in BC patients with different tumor subtypes (ER/PR/HER2/Ki-67), and (2) to consider the patient almost healthy if sdMTSs do not appear during the different diagnostic periods.

Mechanistic Pharmacokinetic/Pharmacodynamic Model of Sunitinib and Dopamine in MCF-7/Adr Xenografts: Linking Cellular Heterogeneity to Tumour Burden

The self-renewal and differentiation of cancer stem-like cells (CSCs) leads to cellular heterogeneity, causing one of the greatest challenges in cancer therapy. Growing evidence suggests that CSC-targeting therapy enhances the effect of concomitant antitumour therapy. To gain an in-depth understanding of this enhanced effect, the kinetic profile of estimated CSC frequency (the fraction of CSCs in tumour) was evaluated for in vivo characterization of cellular heterogeneity using sunitinib and dopamine as a paradigm combination therapy. Female MCF-7/Adr xenografted Balb/c nude mice were treated with sunitinib (p.o., 20 mg/kg) and dopamine (i.p., 50 mg/kg), alone or in combination. Estimated CSC frequency and tumour size were measured over time. Mechanistic PK/PD modelling was performed to quantitatively describe the relationship between drug concentration, estimated CSC frequency and tumour size. Sunitinib reduced tumour size by inducing apoptosis of differentiated tumour cells (DTCs) and enriched CSCs by stimulating its proliferation. Dopamine exhibited anti-CSC effects by suppressing the capacity of CSCs and inducing its differentiation. Simulation and animal studies indicated that concurrent administration was superior to sequential administration under current experimental conditions. Alongside tumour size, the current study provides mechanistic insights into the estimation of CSC frequency as an indicator for cellular heterogeneity. This forms the conceptual basis for in vivo characterization of other combination therapies in preclinical cancer studies.

Engineered 3D Model of Cancer Stem Cell Enrichment and Chemoresistance

Intraperitoneal dissemination of ovarian cancers is preceded by the development of chemoresistant tumors with malignant ascites. Despite the high levels of chemoresistance and relapse observed in ovarian cancers, there are no in vitro models to understand the development of chemoresistance in situ. Method: We describe a highly integrated approach to establish an in vitro model of chemoresistance and stemness in ovarian cancer, using the 3D hanging drop spheroid platform. The model was established by serially passaging non-adherent spheroids. At each passage, the effectiveness of the model was evaluated via measures of proliferation, response to treatment with cisplatin and a novel ALDH1A inhibitor. Concomitantly, the expression and tumor initiating capacity of cancer stem-like cells (CSCs) was analyzed. RNA-seq was used to establish gene signatures associated with the evolution of tumorigenicity, and chemoresistance. Lastly, a mathematical model was developed to predict the emergence of CSCs during serial passaging of ovarian cancer spheroids. Results: Our serial passage model demonstrated increased cellular proliferation, enriched CSCs, and emergence of a platinum resistant phenotype. In vivo tumor xenograft assays indicated that later passage spheroids were significantly more tumorigenic with higher CSCs, compared to early passage spheroids. RNA-seq revealed several gene signatures supporting the emergence of CSCs, chemoresistance, and malignant phenotypes, with links to poor clinical prognosis. Our mathematical model predicted the emergence of CSC populations within serially passaged spheroids, concurring with experimentally observed data. Conclusion: Our integrated approach illustrates the utility of the serial passage spheroid model for examining the emergence and development of chemoresistance in ovarian cancer in a controllable and reproducible format.

Mathematical modeling of bone marrow - peripheral blood dynamics in the disease state based on current emerging paradigms, part II

The cancer stem cell hypothesis has gained currency in recent times but concerns remain about its scientific foundations because of significant gaps that exist between research findings and comprehensive knowledge about cancer stem cells (CSCs). In this light, a mathematical model that considers hematopoietic dynamics in the diseased state of the bone marrow and peripheral blood is proposed and used to address findings about CSCs. The ensuing model, resulting from a modification and refinement of a recent model, develops out of the position that mathematical models of CSC development, that are few at this time, are needed to provide insightful underpinnings for biomedical findings about CSCs as the CSC idea gains traction. Accordingly, the mathematical challenges brought on by the model that mirror general challenges in dealing with nonlinear phenomena are discussed and placed in context. The proposed model describes the logical occurrence of discrete time delays, that by themselves present mathematical challenges, in the evolving cell populations under consideration. Under the challenging circumstances, the steady state properties of the model system of delay differential equations are obtained, analyzed, and the resulting mathematical predictions arising therefrom are interpreted and placed within the framework of findings regarding CSCs. Simulations of the model are carried out by considering various parameter scenarios that reflect different experimental situations involving disease evolution in human hosts. Model analyses and simulations suggest that the emergence of the cancer stem cell population alongside other malignant cells engenders higher dimensions of complexity in the evolution of malignancy in the bone marrow and peripheral blood at the expense of healthy hematopoietic development. The model predicts the evolution of an aberrant environment in which the malignant population particularly in the bone marrow shows tendencies of reaching an uncontrollable equilibrium state. Essentially, the model shows that a structural relationship exists between CSCs and non-stem malignant cells that confers on CSCs the role of temporally enhancing and stimulating the expansion of non-stem malignant cells while also benefitting from increases in their own population and these CSCs may be the main protagonists that drive the ultimate evolution of the uncontrollable equilibrium state of such malignant cells and these may have implications for treatment.

Kinetic modeling of tumor regression incorporating the concept of cancer stem-like cells for patients with locally advanced lung cancer

Background

Personalized medicine for patients receiving radiation therapy remains an elusive goal due, in part, to the limits in our understanding of the underlying mechanisms governing tumor response to radiation. The purpose of this study was to develop a kinetic model, in the context of locally advanced lung cancer, connecting cancer cell subpopulations with tumor volumes measured during the course of radiation treatment for understanding treatment outcome for individual patients.

Methods

The kinetic model consists of three cell compartments: cancer stem-like cells (CSCs), non-stem tumor cells (TCs) and dead cells (DCs). A set of ordinary differential equations were developed to describe the time evolution of each compartment, and the analytic solution of these equations was iterated to be aligned with the day-to-day tumor volume changes during the course of radiation treatment. A least squares fitting method was used to estimate the parameters of the model that include the proportion of CSCs and their radio-sensitivities. This model was applied to five patients with stage III lung cancer, and tumor volumes were measured from 33 cone-beam computed tomography (CBCT) images for each of these patients. The analytical solution of these differential equations was compared with numerically simulated results.

Results

For the five patients with late stage lung cancer, the derived proportions of CSCs are 0.3 on average, the average probability of the symmetry division is 0.057 and the average surviving fractions of CSCs is 0.967, respectively. The derived parameters are comparable to the results from literature and our experiments. The preliminary results suggest that the CSC self-renewal rate is relatively small, compared to the proportion of CSCs for locally advanced lung cancers.

Conclusions

A novel mathematical model has been developed to connect the population of cancer stem-like cells with tumor volumes measured from a sequence of CBCT images. This model may help improve our understanding of tumor response to radiation therapy, and is valuable for development of new treatment regimens for patients with locally advanced lung cancer.

Consolidated mathematical growth model of the primary tumor and secondary distant metastases of breast cancer (CoMPaS)

The goal of this research is to improve the accuracy of predicting the breast cancer (BC) process using the original mathematical model referred to as CoMPaS. The CoMPaS is the original mathematical model and the corresponding software built by modelling the natural history of the primary tumor (PT) and secondary distant metastases (MTS), it reflects the relations between the PT and MTS. The CoMPaS is based on an exponential growth model and consists of a system of determinate nonlinear and linear equations and corresponds to the TNM classification. It allows us to calculate the different growth periods of PT and MTS: 1) a non-visible period for PT, 2) a non-visible period for MTS, and 3) a visible period for MTS. The CoMPaS has been validated using 10-year and 15-year survival clinical data considering tumor stage and PT diameter. The following are calculated by CoMPaS: 1) the number of doublings for the non-visible and visible growth periods of MTS and 2) the tumor volume doubling time (days) for the non-visible and visible growth periods of MTS. The diameters of the PT and secondary distant MTS increased simultaneously. In other words, the non-visible growth period of the secondary distant MTS shrinks, leading to a decrease of the survival of patients with breast cancer. The CoMPaS correctly describes the growth of the PT for patients at the T1aN0M0, T1bN0M0, T1cN0M0, T2N0M0 and T3N0M0 stages, who does not have MTS in the lymph nodes (N0). Additionally, the CoMPaS helps to consider the appearance and evolution period of secondary distant MTS (M1). The CoMPaS correctly describes the growth period of PT corresponding to BC classification (parameter T), the growth period of secondary distant MTS and the 10-15-year survival of BC patients considering the BC stage (parameter M).

A mathematical model for IL-6-mediated, stem cell driven tumor growth and targeted treatment

Targeting key regulators of the cancer stem cell phenotype to overcome their critical influence on tumor growth is a promising new strategy for cancer treatment. Here we present a modeling framework that operates at both the cellular and molecular levels, for investigating IL-6 mediated, cancer stem cell driven tumor growth and targeted treatment with anti-IL6 antibodies. Our immediate goal is to quantify the influence of IL-6 on cancer stem cell self-renewal and survival, and to characterize the subsequent impact on tumor growth dynamics. By including the molecular details of IL-6 binding, we are able to quantify the temporal changes in fractional occupancies of bound receptors and their influence on tumor volume. There is a strong correlation between the model output and experimental data for primary tumor xenografts. We also used the model to predict tumor response to administration of the humanized IL-6R monoclonal antibody, tocilizumab (TCZ), and we found that as little as 1mg/kg of TCZ administered weekly for 7 weeks is sufficient to result in tumor reduction and a sustained deceleration of tumor growth.

A small population of cancer stem cells that share many of the biological characteristics of normal adult stem cells are believed to initiate and sustain tumor growth for a wide variety of malignancies. Growth and survival of these cancer stem cells is highly influenced by tumor micro-environmental factors and molecular signaling initiated by cytokines and growth factors. This work focuses on quantifying the influence of IL-6, a pleiotropic cytokine secreted by a variety of cell types, on cancer stem cell self-renewal and survival. We present a mathematical model for IL-6 mediated, cancer stem cell driven tumor growth that operates at the following levels: (1) the molecular level—capturing cell surface dynamics of receptor-ligand binding and receptor activation that lead to intra-cellular signal transduction cascades; and (2) the cellular level—describing tumor growth, cellular composition, and response to treatments targeted against IL-6.

Formal reasoning about systems biology using theorem proving

System biology provides the basis to understand the behavioral properties of complex biological organisms at different levels of abstraction. Traditionally, analysing systems biology based models of various diseases have been carried out by paper-and-pencil based proofs and simulations. However, these methods cannot provide an accurate analysis, which is a serious drawback for the safety-critical domain of human medicine. In order to overcome these limitations, we propose a framework to formally analyze biological networks and pathways. In particular, we formalize the notion of reaction kinetics in higher-order logic and formally verify some of the commonly used reaction based models of biological networks using the HOL Light theorem prover. Furthermore, we have ported our earlier formalization of Zsyntax, i.e., a deductive language for reasoning about biological networks and pathways, from HOL4 to the HOL Light theorem prover to make it compatible with the above-mentioned formalization of reaction kinetics. To illustrate the usefulness of the proposed framework, we present the formal analysis of three case studies, i.e., the pathway leading to TP53 Phosphorylation, the pathway leading to the death of cancer stem cells and the tumor growth based on cancer stem cells, which is used for the prognosis and future drug designs to treat cancer patients.

Global dynamics of a colorectal cancer treatment model with cancer stem cells

We present and analyze a mathematical model of the treatment of colorectal cancer using a system of nonlinear ordinary differential equations. The model describes the effectiveness of immunotherapy and chemotherapy for treatment of tumor cells and cancer stem cells (CSCs). The effects of CD8⁺T cells, natural killer cells, and interleukin proteins on tumor cells and CSCs under the influence of treatment are also illustrated. Using the method of localization of compact invariant sets, we present conditions on treatment parameters to guarantee a globally attracting tumor clearance state. Numerical simulations using estimated parameters from the literature are included to showcase various global dynamics of the model.

In silico study of the impact of cancer stem cell dynamics and radiobiological hypoxia on tumour response to hyperfractionated radiotherapy

OBJECTIVES

Advanced head and neck carcinomas (HNCs) are aggressive tumours, mainly due to hypoxia and a cancer stem cell (CSC) subpopulation. The aim of this study was to simulate tumour growth and behaviour during radiotherapy of three HNC groups (governed by different growth kinetics, hypoxia levels and CSC division pattern) to determine correlation between resistance factors and responses to hyperfractionated radiotherapy.

METHODS

An in silico HNC model was developed based on biologically realistic input parameters. During radiotherapy simulation, three parameters were studied: growth kinetics, hypoxia and probability of CSC symmetrical division. Both independent and combined effects on tumour response to hyperfractionated radiotherapy were assessed.

RESULTS

Oxic and very mildly hypoxic HNCs were revealed to be controlled by hyperfractionated radiotherapy, irrespective of growth kinetics and CSC division pattern. Moderately hypoxic tumours had different responses to radiotherapy: while slowly proliferating HNCs were still controllable, tumours with higher cell turnover were more resistant. In rapidly proliferating tumours, the number of fractions needed for tumour control increased exponentially with the probability of CSC symmetrical division, whereas in moderately growing HNC, this behaviour was linear. Severely hypoxic tumours could not be controlled by radiotherapy alone. Tumours with CSCs in a severely hypoxic niche required adjuvant therapies to be eradicated.

CONCLUSIONS

Growth kinetics strongly influence tumour responses to treatment. Slowly growing tumours showed linear dependence between dose and hypoxia/CSC, whereas rapidly growing tumours followed exponential behaviour.

The application of information theory for the research of aging and aging-related diseases

This article reviews the application of information-theoretical analysis, employing measures of entropy and mutual information, for the study of aging and aging-related diseases. The research of aging and aging-related diseases is particularly suitable for the application of information theory methods, as aging processes and related diseases are multi-parametric, with continuous parameters coexisting alongside discrete parameters, and with the relations between the parameters being as a rule non-linear. Information theory provides unique analytical capabilities for the solution of such problems, with unique advantages over common linear biostatistics. Among the age-related diseases, information theory has been used in the study of neurodegenerative diseases (particularly using EEG time series for diagnosis and prediction), cancer (particularly for establishing individual and combined cancer biomarkers), diabetes (mainly utilizing mutual information to characterize the diseased and aging states), and heart disease (mainly for the analysis of heart rate variability). Few works have employed information theory for the analysis of general aging processes and frailty, as underlying determinants and possible early preclinical diagnostic measures for aging-related diseases. Generally, the use of information-theoretical analysis permits not only establishing the (non-linear) correlations between diagnostic or therapeutic parameters of interest, but may also provide a theoretical insight into the nature of aging and related diseases by establishing the measures of variability, adaptation, regulation or homeostasis, within a system of interest. It may be hoped that the increased use of such measures in research may considerably increase diagnostic and therapeutic capabilities and the fundamental theoretical mathematical understanding of aging and disease.

Enabling personalized cancer medicine decisions: The challenging pharmacological approach of PBPK models for nanomedicine and pharmacogenomics (Review)

The existing tumor heterogeneity and the complexity of cancer cell biology critically demand powerful translational tools with which to support interdisciplinary efforts aiming to advance personalized cancer medicine decisions in drug development and clinical practice. The development of physiologically based pharmacokinetic (PBPK) models to predict the effects of drugs in the body facilitates the clinical translation of genomic knowledge and the implementation of in vivo pharmacology experience with pharmacogenomics. Such a direction unequivocally empowers our capacity to also make personalized drug dosage scheme decisions for drugs, including molecularly targeted agents and innovative nanoformulations, i.e. in establishing pharmacotyping in prescription. In this way, the applicability of PBPK models to guide individualized cancer therapeutic decisions of broad clinical utility in nanomedicine in real-time and in a cost-affordable manner will be discussed. The latter will be presented by emphasizing the need for combined efforts within the scientific borderlines of genomics with nanotechnology to ensure major benefits and productivity for nanomedicine and personalized medicine interventions.

A Branching Process to Characterize the Dynamics of Stem Cell Differentiation

The understanding of the regulatory processes that orchestrate stem cell maintenance is a cornerstone in developmental biology. Here, we present a mathematical model based on a branching process formalism that predicts average rates of proliferative and differentiative divisions in a given stem cell population. In the context of vertebrate neurogenesis, the model predicts complex non-monotonic variations in the rates of pp, pd and dd modes of division as well as in cell cycle length, in agreement with experimental results. Moreover, the model shows that the differentiation probability follows a binomial distribution, allowing us to develop equations to predict the rates of each mode of division. A phenomenological simulation of the developing spinal cord informed with the average cell cycle length and division rates predicted by the mathematical model reproduces the correct dynamics of proliferation and differentiation in terms of average numbers of progenitors and differentiated cells. Overall, the present mathematical framework represents a powerful tool to unveil the changes in the rate and mode of division of a given stem cell pool by simply quantifying numbers of cells at different times.

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

Background

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

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

Results

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

Conclusions

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

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

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

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

Laboratory models for central nervous system tumor stem cell research

Central nervous system (CNS) tumors are complex organ systems comprising of a neoplastic component with associated vasculature, inflammatory cells, and reactive cellular and extracellular components. Research has identified a subset of cells in CNS tumors that portray defining properties of neural stem cells, namely, that of self-renewal and multi-potency. Growing evidence suggests that these tumor stem cells (TSC) play an important role in the maintenance and growth of the tumor. Furthermore, these cells have also been shown to be refractory to conventional therapy and may be crucial for tumor recurrence and metastasis. Current investigations are focusing on isolating these TSC from CNS tumors to investigate their unique biological processes. This understanding will help identify and develop more effective and comprehensive treatment strategies. This chapter provides an overview of some of the most commonly used laboratory models for CNSTSC research.

Concepts in solid tumor evolution

Evolutionary mechanisms in cancer progression give tumors their individuality. Cancer evolution is different from organismal evolution, however, and we discuss where concepts from evolutionary genetics are useful or limited in facilitating an understanding of cancer. Based on these concepts we construct and apply the simplest plausible model of tumor growth and progression. Simulations using this simple model illustrate the importance of stochastic events early in tumorigenesis, highlight the dominance of exponential growth over linear growth and differentiation, and explain the clonal substructure of tumors.

What Tumor Dynamics Modeling Can Teach us About Exploiting the Stem-Cell View for Better Cancer Treatment

The cancer stem cell hypothesis is that in human solid cancers, only a small proportion of the cells, the cancer stem cells (CSCs), are self-renewing; the vast majority of the cancer cells are unable to sustain tumor growth indefinitely on their own. In recent years, discoveries have led to the concentration, if not isolation, of putative CSCs. The evidence has mounted that CSCs do exist and are important. This knowledge may promote better understanding of treatment resistance, create opportunities to test agents against CSCs, and open up promise for a fresh approach to cancer treatment. The first clinical trials of new anti-CSC agents are completed, and many others follow. Excitement is mounting that this knowledge will lead to major improvements, even breakthroughs, in treating cancer. However, exploitation of this phenomenon may be more successful if informed by insights into the population dynamics of tumor development. We revive some ideas in tumor dynamics modeling to extract some guidance in designing anti-CSC treatment regimens and the clinical trials that test them.

Numerical resolution of a model of tumour growth

We consider and solve numerically a mathematical model of tumour growth based on cancer stem cells (CSC) hypothesis with the aim of gaining some insight into the relation of different processes leading to exponential growth in solid tumours and into the evolution of different subpopulations of cells. The model consists of four hyperbolic equations of first order to describe the evolution of four subpopulations of cells. A fifth equation is introduced to model the evolution of the moving boundary. The coefficients of the model represent the rates at which reactions occur. In order to integrate numerically the four hyperbolic equations, a formulation in terms of the total derivatives is posed. A finite element discretization is applied to integrate the model equations in space. Our numerical results suggest the existence of a pseudo-equilibrium state reached at the early stage of the tumour, for which the fraction of CSC remains small. We include the study of the behaviour of the solutions for longer times and we obtain that the solutions to the system of partial differential equations stabilize to homogeneous steady states whose values depend only on the values of the parameters. We show that CSC may comprise different proportions of the tumour, becoming, in some cases, the predominant type of cells within the tumour. We also obtain that possible effective measure to detain tumour progression should combine the targeting of CSC with the targeting of progenitor cells.

Cancer stem cells: small subpopulation or evolving fraction?

This review discusses quantitative modeling studies of stem and non-stem cancer cell interactions and the fraction of cancer stem cells.

A mathematical-biological joint effort to investigate the tumor-initiating ability of Cancer Stem Cells

The involvement of Cancer Stem Cells (CSCs) in tumor progression and tumor recurrence is one of the most studied subjects in current cancer research. The CSC hypothesis states that cancer cell populations are characterized by a hierarchical structure that affects cancer progression. Due to the complex dynamics involving CSCs and the other cancer cell subpopulations, a robust theory explaining their action has not been established yet. Some indications can be obtained by combining mathematical modeling and experimental data to understand tumor dynamics and to generate new experimental hypotheses. Here, we present a model describing the initial phase of ErbB2⁺ mammary cancer progression, which arises from a joint effort combing mathematical modeling and cancer biology. The proposed model represents a new approach to investigate the CSC-driven tumorigenesis and to analyze the relations among crucial events involving cancer cell subpopulations. Using in vivo and in vitro data we tuned the model to reproduce the initial dynamics of cancer growth, and we used its solution to characterize observed cancer progression with respect to mutual CSC and progenitor cell variation. The model was also used to investigate which association occurs among cell phenotypes when specific cell markers are considered. Finally, we found various correlations among model parameters which cannot be directly inferred from the available biological data and these dependencies were used to characterize the dynamics of cancer subpopulations during the initial phase of ErbB2⁺ mammary cancer progression.

Targeting cancer stem cell plasticity through modulation of epidermal growth factor and insulin-like growth factor receptor signaling in head and neck squamous cell cancer

Emerging data suggest that cancer stem cells (CSCs) exist in equilibrium with differentiated cells and that stochastic transitions between these states can account for tumor heterogeneity and drug resistance. The aim of this study was to establish an in vitro system that recapitulates stem cell plasticity in head and neck squamous cell cancers (HNSCCs) and identify the factors that play a role in the maintenance and repopulation of CSCs. Tumor spheres were established using patient-derived cell lines via anchorage-independent cell culture techniques. These tumor spheres were found to have higher aldehyde dehydrogenase (ALD) cell fractions and increased expression of Kruppel-like factor 4, SRY (sex determining region Y)-box 2, and Nanog and were resistant to γ-radiation, 5-fluorouracil, cisplatin, and etoposide treatment compared with monolayer culture cells. Monolayer cultures were subject to single cell cloning to generate clones with high and low ALD fractions. ALDHigh clones showed higher expression of stem cell and epithelial-mesenchymal transition markers compared with ALDLow clones. ALD fractions, representing stem cell fractions, fluctuated with serial passaging, equilibrating at a level specific to each cell line, and could be augmented by the addition of epidermal growth factor (EGF) and/or insulin. ALDHigh clones showed increased EGF receptor (EGFR) and insulin-like growth factor-1 receptor (IGF-1R) phosphorylation, with increased activation of downstream pathways compared with ALDLow clones. Importantly, blocking these pathways using specific inhibitors against EGFR and IGF-1R reduced stem cell fractions drastically. Taken together, these results show that HNSCC CSCs exhibit plasticity, with the maintenance of the stem cell fraction dependent on the EGFR and IGF-1R pathways and potentially amenable to targeted therapeutics.

A multicompartment mathematical model of cancer stem cell-driven tumor growth dynamics

Tumors are appreciated to be an intrinsically heterogeneous population of cells with varying proliferation capacities and tumorigenic potentials. As a central tenet of the so-called cancer stem cell hypothesis, most cancer cells have only a limited lifespan, and thus cannot initiate or reinitiate tumors. Longevity and clonogenicity are properties unique to the subpopulation of cancer stem cells. To understand the implications of the population structure suggested by this hypothesis--a hierarchy consisting of cancer stem cells and progeny non-stem cancer cells which experience a reduction in their remaining proliferation capacity per division--we set out to develop a mathematical model for the development of the aggregate population. We show that overall tumor progression rate during the exponential growth phase is identical to the growth rate of the cancer stem cell compartment. Tumors with identical stem cell proportions, however, can have different growth rates, dependent on the proliferation kinetics of all participating cell populations. Analysis of the model revealed that the proliferation potential of non-stem cancer cells is likely to be small to reproduce biologic observations. Furthermore, a single compartment of non-stem cancer cell population may adequately represent population growth dynamics only when the compartment proliferation rate is scaled with the generational hierarchy depth.

A multi-phenotypic cancer model with cell plasticity

The conventional cancer stem cell (CSC) theory indicates a hierarchy of CSCs and non-stem cancer cells (NSCCs), that is, CSCs can differentiate into NSCCs but not vice versa. However, an alternative paradigm of CSC theory with reversible cell plasticity among cancer cells has received much attention very recently. Here we present a generalized multi-phenotypic cancer model by integrating cell plasticity with the conventional hierarchical structure of cancer cells. We prove that under very weak assumption, the nonlinear dynamics of multi-phenotypic proportions in our model has only one stable steady state and no stable limit cycle. This result theoretically explains the phenotypic equilibrium phenomena reported in various cancer cell lines. Furthermore, according to the transient analysis of our model, it is found that cancer cell plasticity plays an essential role in maintaining the phenotypic diversity in cancer especially during the transient dynamics. Two biological examples with experimental data show that the phenotypic conversions from NCSSs to CSCs greatly contribute to the transient growth of CSCs proportion shortly after the drastic reduction of it. In particular, an interesting overshooting phenomenon of CSCs proportion arises in three-phenotypic example. Our work may pave the way for modeling and analyzing the multi-phenotypic cell population dynamics with cell plasticity.

A model of hematopoietic stem cell proliferation under the influence of a chemotherapeutic agent in combination with a hematopoietic inducing agent

BACKGROUND

Hematopoiesis is a complex process that encompasses both pro-mitotic and anti-mitotic stimuli. Pharmacological agents used in chemotherapy have a prominent anti-mitotic effect. The approach of inhibiting cell proliferation is rational with respect to the rapidly dividing malignant cells. However, it poses a serious problem with respect to cell proliferation of cell types required for the 'house-keeping' operations of the human body. One such affected system is hematopoiesis. Chemotherapy induced anemia is an undesired side effect of chemotherapy that can lead to serious complications. Patients exhibiting anemia or leukopenia during chemotherapy are frequently administered a hematopoietic inducing agent that enhances hematopoiesis.

METHODS

In previous work, we derived a mathematical model consisting of a set of delay differential equations that was dependent on the effect of a hematopoietic inducing agent. The aim of the current work was to formulate a mathematical model that captures both the effect of a chemotherapeutic agent in combination with a hematopoietic inducing agent. Steady state solutions and stability analysis of the system of equations is performed and numerical simulations of the stem cell population are provided.

RESULTS

Numerical simulations confirm that our mathematical model captures the desired result which is that the use of hematopoietic agents in conjunction with chemotherapeutic agents can decrease the negative secondary effects often experienced by patients.

CONCLUSIONS

The proposed model indicates that the introduction of hematopoietic inducing agents have clinical potential to offset the deleterious effects of chemotherapy treatment. Furthermore, the proposed model is relevant in that it enhances the understanding of stem cell dynamics and provides insight on the stem cell kinetics.

Dynamics of tissue topology during cancer invasion and metastasis

During tumor progression, cancer cells mix with other cell populations including epithelial and endothelial cells. Although potentially important clinically as well as for our understanding of basic tumor biology, the process of mixing is largely a mystery. Furthermore, there is no rigorous, analytical measure available for quantifying the mixing of compartments within a tumor. I present here a mathematical model of tissue repair and tumor growth based on collective cell migration that simulates a wide range of observed tumor behaviors with correct tissue compartmentalization and connectivity. The resulting dynamics are analyzed in light of the Euler characteristic number (χ), which describes key topological features such as fragmentation, looping and cavities. The analysis predicts a number of regimes in which the cancer cells can encapsulate normal tissue, form a co-interdigitating mass, or become fragmented and encapsulated by endothelial or epithelial structures. Key processes that affect the topological changes are the production of provisional matrix in the tumor, and the migration of endothelial or epithelial cells on this matrix. Furthermore, the simulations predict that topological changes during tumor invasion into blood vessels may contribute to metastasis. The topological analysis outlined here could be useful for tumor diagnosis or monitoring response to therapy and would only require high resolution, 3D image data to resolve and track the various cell compartments.

A mathematical model of cancer stem cell driven tumor initiation: implications of niche size and loss of homeostatic regulatory mechanisms

Hierarchical organized tissue structures, with stem cell driven cell differentiation, are critical to the homeostatic maintenance of most tissues, and this underlying cellular architecture is potentially a critical player in the development of a many cancers. Here, we develop a mathematical model of mutation acquisition to investigate how deregulation of the mechanisms preserving stem cell homeostasis contributes to tumor initiation. A novel feature of the model is the inclusion of both extrinsic and intrinsic chemical signaling and interaction with the niche to control stem cell self-renewal. We use the model to simulate the effects of a variety of types and sequences of mutations and then compare and contrast all mutation pathways in order to determine which ones generate cancer cells fastest. The model predicts that the sequence in which mutations occur significantly affects the pace of tumorigenesis. In addition, tumor composition varies for different mutation pathways, so that some sequences generate tumors that are dominated by cancerous cells with all possible mutations, while others are primarily comprised of cells that more closely resemble normal cells with only one or two mutations. We are also able to show that, under certain circumstances, healthy stem cells diminish due to the displacement by mutated cells that have a competitive advantage in the niche. Finally, in the event that all homeostatic regulation is lost, exponential growth of the cancer population occurs in addition to the depletion of normal cells. This model helps to advance our understanding of how mutation acquisition affects mechanisms that influence cell-fate decisions and leads to the initiation of cancers.

Population dynamics of cancer cells with cell state conversions

Cancer stem cell (CSC) theory suggests a cell-lineage structure in tumor cells in which CSCs are capable of giving rise to the other non-stem cancer cells (NSCCs) but not vice versa. However, an alternative scenario of bidirectional interconversions between CSCs and NSCCs was proposed very recently. Here we present a general population model of cancer cells by integrating conventional cell divisions with direct conversions between different cell states, namely, not only can CSCs differentiate into NSCCs by asymmetric cell division, NSCCs can also dedifferentiate into CSCs by cell state conversion. Our theoretical model is validated when applying the model to recent experimental data. It is also found that the transient increase in CSCs proportion initiated from the purified NSCCs subpopulation cannot be well predicted by the conventional CSC model where the conversion from NSCCs to CSCs is forbidden, implying that the cell state conversion is required especially for the transient dynamics. The theoretical analysis also gives the condition such that our general model can be equivalently reduced into a simple Markov chain with only cell state transitions keeping the same cell proportion dynamics.

Cancer Stem Cells: A Minor Cancer Subpopulation that Redefines Global Cancer Features

In recent years cancer stem cells (CSCs) have been hypothesized to comprise only a minor subpopulation in solid tumors that drives tumor initiation, progression, and metastasis; the so-called “cancer stem cell hypothesis.” While a seemingly trivial statement about numbers, much is put at stake. If true, the conclusions of many studies of cancer cell populations could be challenged, as the bulk assay methods upon which they depend have, by, and large, taken for granted the notion that a “typical” cell of the population possesses the attributes of a cell capable of perpetuating the cancer, i.e., a CSC. In support of the CSC hypothesis, populations enriched for so-called “tumor-initiating” cells have demonstrated a corresponding increase in tumorigenicity as measured by dilution assay, although estimates have varied widely as to what the fractional contribution of tumor-initiating cells is in any given population. Some have taken this variability to suggest the CSC fraction may be nearly 100% after all, countering the CSC hypothesis, and that there are simply assay-dependent error rates in our ability to “reconfirm” CSC status at the cell level. To explore this controversy more quantitatively, we developed a simple cellular automaton model of CSC-driven tumor growth dynamics. Assuming CSC and non-stem cancer cells (CC) subpopulations coexist to some degree, we evaluated the impact of an environmentally dependent CSC symmetric division probability and a CC proliferation capacity on tumor progression and morphology. Our model predicts, as expected, that the frequency of CSC divisions that are symmetric highly influences the frequency of CSCs in the population, but goes on to predict the two frequencies can be widely divergent, and that spatial constraints will tend to increase the CSC fraction over time. Further, tumor progression times show a marked dependence on both the frequency of CSC divisions that are symmetric and on the proliferation capacities of CC. Together, these findings can explain, within the CSC hypothesis, the widely varying measures of stem cell fractions observed. In particular, although the CSC fraction is influenced by the (environmentally modifiable) CSC symmetric division probability, with the former converging to unity as the latter nears 100%, the CSC fraction becomes quite small even for symmetric division probabilities modestly lower than 100%. In the latter case, the tumor exhibits a clustered morphology and the CSC fraction steadily increases with time; more so on both counts when the death rate of CCs is higher. Such variations in CSC fraction and morphology are not only consistent with the CSC hypothesis, but lend support to it as one expected byproduct of the dynamical interactions that are predicted to take place among a relatively small CSC population, its CC counterpart, and the host compartment over time.