Mathematical model for the cancer stem cell hypothesis

Recent research on the origin of brain cancer has implicated a subpopulation of self-renewing brain cancer stem cells for malignant tumour growth. Various genes that regulate self-renewal in normal stem cells are also found in cancer stem cells. This implies that cancers can occur because of mutations in normal stem cells and early progenitor cells. A predictive mathematical model based on the cell compartment method is presented here to pose and validate non-intuitive scenarios proposed through the neural cancer stem cell hypothesis. The growths of abnormal (stem and early progenitor) cells from their normal counterparts are ascribed with separate mutation probabilities. Stem cell mutations are found to be more significant for the development of cancer than a similar mutation in the early progenitor cells. The model also predicts that, as previously hypothesized, repeated insult to mature cells increases the formation of abnormal progeny, and hence the risk of cancer.     

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

Background

The Cancer Stem Cell (CSC) hypothesis has gained credibility within the cancer research community. According to this hypothesis, a small subpopulation of cells within cancerous tissues exhibits stem-cell-like characteristics and is responsible for the maintenance and proliferation of cancer.

Methodologies/Principal Findings

We present a simple compartmental pseudo-chemical mathematical model for tumor growth, based on the CSC hypothesis, and derived using a “chemical reaction” approach. We defined three cell subpopulations: CSCs, transit progenitor cells, and differentiated cells. Each event related to cell division, differentiation, or death is then modeled as a chemical reaction. The resulting set of ordinary differential equations was numerically integrated to describe the time evolution of each cell subpopulation and the overall tumor growth. The parameter space was explored to identify combinations of parameter values that produce biologically feasible and consistent scenarios.

Conclusions/Significance

Certain kinetic relationships apparently must be satisfied to sustain solid tumor growth and to maintain an approximate constant fraction of CSCs in the tumor lower than 0.01 (as experimentally observed): (a) the rate of symmetrical and asymmetrical CSC renewal must be in the same order of magnitude; (b) the intrinsic rate of renewal and differentiation of progenitor cells must be half an order of magnitude higher than the corresponding intrinsic rates for cancer stem cells; (c) the rates of apoptosis of the CSC, transit amplifying progenitor (P) cells, and terminally differentiated (D) cells must be progressively higher by approximately one order of magnitude. Simulation results were consistent with reports that have suggested that encouraging CSC differentiation could be an effective therapeutic strategy for fighting cancer in addition to selective killing or inhibition of symmetric division of CSCs.     

A unified model of sigmoid tumour growth based on cell proliferation and quiescence

Abstract.   Objectives : A class of sigmoid functions designated generalized von Bertalanffy, Gompertzian and generalized Logistic has been used to fit tumour growth data. Various models have been proposed to explain the biological significance and foundations of these functions. However, no model has been found to fully explain all three or the relationships between them. Materials and Methods : We propose a simple cancer cell population dynamics model that provides a biological interpretation for these sigmoids' ability to represent tumour growth. Results and Conclusions : We show that the three sigmoids can be derived from the model and are in fact a single solution subject to the continuous variation of parameters describing the decay of the proliferation fraction and/or cell quiescence. We use the model to generate proliferation fraction profiles for each sigmoid and comment on the significance of the differences relative to cell cycle-specific and non-cell cycle-specific therapies.     

Mathematical model for human myeloma relating growth kinetics and drug resistance

We present a computer-based mathematical model that can simulate characteristic features of the clinical time course of human myeloma. It asserts that therapy resistance in myeloma cells is an inherited trait associated with the longer inter-mitotic times of some cells and that the strength of this trait affects tumour growth characteristics. These kinetic differences within the malignant cell clone may also influence therapeutic efficacy. In the model, the same total therapy, administered in different time-dose fractions, could be 'curative' or 'minimally effective' depending on kinetic properties. For example, as others have shown, in myeloma pulsed intermittent therapy is often more effective than low dose continuous therapy. According to our model this finding is compatible with a high coefficient of inheritability of resistance from one cell generation to the next. The model also suggests that if there are subclones of varying resistance, a therapy must have some effect on each of them if it is to be employed in a curative fashion. While many aspects of the model are not yet clinically testable, exploration of its concepts might increase knowledge about fundamental neoplastic mechanisms.     

Mathematical analysis of cancer chemotherapy: The effects of chemotherapeutic agents on the cell cycle traverse

For the tumor model of Skipper and Zubrod, which has been analyzed previously for the theoretical FLM function and the effect of chemotherapy against tumors of known or assumed kinetic characteristics, the theoretical continuous labeling (CL) function is derived by considering an equivalent tumor (in terms of unlabeled cell populations) in which the density function of phase duration of cells inS-phasef ₂(a ₂)=δ(a ₂−∞) and the loss functionL ₂(t)=0. This mathematical concept of blocking is applied to the analysis of synchronization in tumor growth and blocking effects in cancer chemotherapy. These effects of chemical agents on the cell cycle progression are being incorporated into a previously written computer simulation program for cancer chemotherapy. Whereas, a program is written and used to simulate the CL functions for L1210 leukemia, and primary and metastatic Lewis lung carcinoma.     

The stem cell factor antibody enhances the chemotherapeutic effect of adriamycin on chemoresistant breast cancer cells

Background

The outcome of chemotherapy in breast cancer is strongly influenced by multidrug resistance (MDR). Several surrogate markers of chemoresistance have been identified including - CD24 (cluster differentiation 24) expression, stem cell growth factor (SCF), B-cell lymphocyte protein 2 (Bcl-2) and annexin V. The present study aimed to examine the expression of CD24 in the sensitive breast cancer cell line MCF-7 (Michigan Foudation-7) and MCF-7/adriamycin resistant (MCF-7/AdrRes) cells, and, if minimal effective doses of the anthracycline drug adriamycin (0.579???M and 88.2???M) would be enhanced by the antibody to SCF (anti-SCF).

Methods

CD24 expression was analysed by flow cytometry. Both Bcl-2 and annexin V protein expression were quantitatively assessed by the enzyme-linked immunosorbent assay (ELISA).

Results

In MCF-7/AdrRes cells the expression of CD24 was significantly higher compared to MCF-7 cells, 86.6% and 16.3% (p?Conclusion Adding anti-SCF to the chemotherapeutic regime of adriamycin may strongly enhance its chemotherapeutic effect in the treatment of patients with breast cancer.     

Evolution of cell motility in an individual-based model of tumour growth

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.     

Poly-2'-DNP-RNAs with enhanced efficacy for inhibiting cancer cell growth

It is often believed that small interfering RNA (siRNA) is at least 10-fold more effective than the single-stranded antisense oligonucleotide for silencing the same target gene in the same cells. In view of the recent discovery that the RNA-induced silencing complex (RISC) contains only a single-stranded RNA (ssRNA) molecule and can be reconstituted using single-stranded antisense RNA, such a large difference in efficacy seems puzzling. One possible reason is that hybridization protects siRNA from hydrolysis by endogenous RNase activity until it is incorporated in the RISC, whereas ssRNA is rapidly hydrolyzed. Because the single-stranded poly-2'-O-(2,4-dinitrophenyl)-RNA (DNP-ssRNA) is both RNase resistant and membrane permeable, we synthesized homologous native siRNAs, DNP-siRNAs, native ssRNAs, and DNP-ssRNAs and made a comparative study of their efficacies for inhibiting the growth of two cancer cell lines with different overexpressed target genes under equivalent experimental conditions. It was found that the efficacy of antisense DNP-ssRNA is higher than that of the corresponding siRNA and that the efficacy of native siRNA for inhibiting cell growth can also be enhanced from 2-fold to 6-fold by replacing the native strands of RNA in siRNA with homologous DNP-RNA. Thermal denaturation data show that the hybridization affinity of the DNP-RNA/RNA duplex is higher than that of the native RNA/RNA duplex. Western blotting analysis of A549 cells treated with antisense DNP-ssRNAs containing single mismatching bases shows that the gene silencing by antisense DNP-ssRNA is as sequence specific as that by siRNA. The observed large enhancement of inhibition efficacy of native RNAs by DNP derivatization should be advantageous for both gene silencing studies and therapeutic applications.     

Modelling cell population growth with applications to cancer therapy in human tumour cell lines

In this paper we present an overview of the work undertaken to model a population of cells and the effects of cancer therapy. We began with a theoretical one compartment size structured cell population model and investigated its asymptotic steady size distributions (SSDs) (On a cell growth model for plankton, MMB JIMA 21 (2004) 49). However these size distributions are not similar to the DNA (size) distributions obtained experimentally via the flow cytometric analysis of human tumour cell lines (data obtained from the Auckland Cancer Society Research Centre, New Zealand). In our one compartment model, size was a generic term, but in order to obtain realistic steady size distributions we chose size to be DNA content and devised a multi-compartment mathematical model for the cell division cycle where each compartment corresponds to a distinct phase of the cell cycle (J. Math. Biol. 47 (2003) 295). We then incorporated another compartment describing the possible induction of apoptosis (cell death) from mitosis phase (Modelling cell death in human tumour cell lines exposed to anticancer drug paclitaxel, J. Math. Biol. 2004, in press). This enabled us to compare our model to flow cytometric data of a melanoma cell line where the anticancer drug, paclitaxel, had been added. The model gives a dynamic picture of the effects of paclitaxel on the cell cycle. We hope to use the model to describe the effects of other cancer therapies on a number of different cell lines.     

On the proportion of cancer stem cells in a tumour

It is now generally accepted that cancers contain a sub-population, the cancer stem cells (CSCs), which initiate and drive a tumour’s growth. At least until recently it has been widely assumed that only a small proportion of the cells in a tumour are CSCs. Here we use a mathematical model, supported by experimental evidence, to show that such an assumption is unwarranted. We show that CSCs may comprise any possible proportion of the tumour, and that the higher the proportion the more aggressive the tumour is likely to be.     

The small intestine as a model for evaluating adult tissue stem cell drug targets

Adult tissue stem cells are defined and some current controversies are discussed. These crucial cells are responsible for all cell production in renewing tissues, and play a vital role in tissue regeneration. Although reliable stem cell markers are generally unavailable for adult epithelial tissues, the small intestinal crypts are an excellent in vivo model system to study stem cells. Within this tissue, the stem cells have a very well-defined cell position, allowing accurate definition of stem cell specific events. Clonal regeneration assays for the small intestine allow stem cell survival and functional competence to be studied. The ultimate lineage ancestor stem cells are extremely efficiently protected from genetic damage, which accounts for the low cancer incidence in this tissue. Some of the regulatory networks governing stem and transit cell behaviour are beginning to be understood and it is postulated that p53 plays a crucial role in these processes.     

A cellular automaton model for tumour growth in inhomogeneous environment

Most of the existing mathematical models for tumour growth and tumour-induced angiogenesis neglect blood flow. This is an important factor on which both nutrient and metabolite supply depend. In this paper we aim to address this shortcoming by developing a mathematical model which shows how blood flow and red blood cell heterogeneity influence the growth of systems of normal and cancerous cells. The model is developed in two stages. First we determine the distribution of oxygen in a native vascular network, incorporating into our model features of blood flow and vascular dynamics such as structural adaptation, complex rheology and red blood cell circulation. Once we have calculated the oxygen distribution, we then study the dynamics of a colony of normal and cancerous cells, placed in such a heterogeneous environment. During this second stage, we assume that the vascular network does not evolve and is independent of the dynamics of the surrounding tissue. The cells are considered as elements of a cellular automaton, whose evolution rules are inspired by the different behaviour of normal and cancer cells. Our aim is to show that blood flow and red blood cell heterogeneity play major roles in the development of such colonies, even when the red blood cells are flowing through the vasculature of normal, healthy tissue.     

Axitinib targeted cancer stemlike cells to enhance efficacy of chemotherapeutic drugs via inhibiting the drug transport function of ABCG2

Stemlike cells have been isolated by their ability to efflux Hoechst 33342 dye and are called the side population (SP). We evaluated the effect of axitinib on targeting cancer stemlike cells and enhancing the efficacy of chemotherapeutical agents. We found that axitinib enhanced the cytotoxicity of topotecan and mitoxantrone in SP cells sorted from human lung cancer A549 cells and increased cell apoptosis induced by chemotherapeutical agents. Moreover, axitinib particularly inhibited the function of adenosine triphosphate (ATP)-binding cassette subfamily G member 2 (ABCG2) and reversed ABCG2-mediated multidrug resistance (MDR) in vitro. However, no significant reversal effect was observed in ABCB1-, ABCC1- or lung resistance-related protein (LRP)-mediated MDR. Furthermore, in both sensitive and MDR cancer cells axitinib neither altered the expression of ABCG2 at the mRNA or protein levels nor blocked the phosphorylation of AKT and extracellular signal-regulated kinase (ERK)1/2. In nude mice bearing ABCG2-overexpressing S1-M1-80 xenografts, axitinib significantly enhanced the antitumor activity of topotecan without causing additional toxicity. Taken together, these data suggest that axitinib particularly targets cancer stemlike cells and reverses ABCG2-mediated drug resistance by inhibiting the transporter activity of ABCG2.     

Accounting for quiescent cells in tumour growth and cancer treatment

A four-state cell-cycle model with explicit G1-phase representation, termed the quiescent-cell model (QCM), has been proposed to represent biologically the G1-phase specific effect of the chemotherapeutic tamoxifen. The QCM was used to model untreated and tamoxifen-treated tumour xenograft data from the literature with equivalent accuracy to previously developed tumour growth models. Open-loop analysis demonstrated that perturbations to the two newly introduced parameters, kG01 and kG10, significantly altered untreated tumour growth predictions. However, the sensitivity did not carry over to closed-loop simulations, where alterations to kD and kGS proved most significant in determining overall controller performance. Additional mismatch studies comparing controllers designed using the QCM to controllers designed with the Gompertz model and saturating-rate, cell-cycle model returned similar performance for a step-wise tumour reduction case study, but the quiescent-cell controller delivered a more aggressive treatment regimen. More importantly, the Gompertz and saturating-rate, cell-cycle controllers were unable to follow a reference trajectory when measurement updates were made biweekly, with both controllers returning tamoxifen dose schedules alternating between the maximum and minimum allowable dose.     

Prediction of drug efficacy for cancer treatment based on comparative analysis of chemosensitivity and gene expression data

The NCI60 database is the largest available collection of compounds with measured anti-cancer activity. The strengths and limitations for using the NCI60 database as a source of new anti-cancer agents are explored and discussed in relation to previous studies. We selected a sub-set of 2333 compounds with reliable experimental half maximum growth inhibitions (GI(50)) values for 30 cell lines from the NCI60 data set and evaluated their growth inhibitory effect (chemosensitivity) with respect to tissue of origin. This was done by identifying natural clusters in the chemosensitivity data set and in a data set of expression profiles of 1901 genes for the corresponding tumor cell lines. Five clusters were identified based on the gene expression data using self-organizing maps (SOM), comprising leukemia, melanoma, ovarian and prostate, basal breast, and luminal breast cancer cells, respectively. The strong difference in gene expression between basal and luminal breast cancer cells was reflected clearly in the chemosensitivity data. Although most compounds in the data set were of low potency, high efficacy compounds that showed specificity with respect to tissue of origin could be found. Furthermore, eight potential topoisomerase II inhibitors were identified using a structural similarity search. Finally, a set of genes with expression profiles that were significantly correlated with anti-cancer drug activity was identified. Our study demonstrates that the combined data sets, which provide comprehensive information on drug activity and gene expression profiles of tumor cell lines studied, are useful for identifying potential new active compounds.     

Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites

Poor drug delivery and low rates of cell proliferation are two factors associated with hypoxia that diminish the efficacy of many chemotherapeutic drugs. Since macrophages are known to migrate specifically towards, and localize within, hypoxic tumour regions, a promising resolution to these problems involves genetically engineering macrophages to perform such anti-tumour functions as inducing cell lysis and inhibiting angiogenesis. In this paper we outline a modelling approach to characterize macrophage infiltration into early avascular solid tumours, and extensions to study the interaction of these cells with macrophages already present within the tumour. We investigate the role of chemotaxis and chemokine production, and the efficacy of macrophages as vehicles for drug delivery to hypoxic tumour sites. The model is based upon a growing avascular tumour spheroid, in which volume is filled by tumour cells, macrophages and extracellular material, and tumour cell proliferation and death is regulated by nutrient diffusion. Crucially, macrophages occupy volume, and hence contribute to the volume balance and hence the size of the tumour. We also include oxygen-dependent production of macrophage chemokines, which can lead to accumulations in the hypoxic region of the tumour. We find that the macrophage chemotactic sensitivity is a key determinant of macrophage infiltration and tumour size. Although increased infiltration should be beneficial from the point of view of macrophage-based therapies, such infiltration in fact leads to increased tumour sizes. Finally, we include terms representing the induced death of tumour cells by hypoxic engineered macrophages. We demonstrate that reductions in tumour size can be achieved, but predict that a combination of therapies would be required for complete eradication. We also highlight some counter-intuitive predictions-for example, absolute and relative measures of tumour burden lead to different conclusions about prognosis. In summary, this paper illustrates how mathematical models may be used to investigate promising macrophage-based therapies.     

Mathematical model of human growth hormone (hGH)-stimulated cell proliferation explains the efficacy of hGH variants as receptor agonists or antagonists

Human growth hormone (hGH) is a therapeutically important endocrine factor that signals various cell types. Structurally and functionally, the interactions of hGH with its receptor have been resolved in fine detail, such that hGH and hGH receptor variants can be practically engineered by either random or rational approaches to achieve significant changes in the free energies of binding. A somewhat unique feature of hGH action is its homodimerization of two hGH receptors, which is required for intracellular signaling and stimulation of cell proliferation, yet the potencies of hGH mutants in cell-based assays rarely correlate with their overall receptor-binding avidities. Here, a mathematical model of hGH-stimulated cell signaling is posed, accounting not only for binding interactions at the cell surface but induction of receptor endocytosis and downregulation as well. Receptor internalization affects ligand potency by imposing a limit on the lifetime of an active receptor complex, irrespective of ligand-receptor binding properties. The model thus explains, in quantitative terms, the numerous published observations regarding hGH receptor agonism and antagonism and challenges the interpretations of previous studies that have not considered receptor trafficking as a central regulatory mechanism in hGH signaling.     

A conformation hypothesis for the suppressor and promoter functions of p53 in cell growth control and in cancer.

Cancer is a genetic disease caused by defective control of cell proliferation. As cancer cells divide, the genetic defect is inherited by each daughter cell, leading to tumour development with possible progression to malignancy. The identification of those genes linked with cancer is essential for our understanding of the regulation of cell proliferation and for the therapeutic management of cancer cell growth. Recent studies have revealed that p53 is the most commonly affected gene in human cancer. It is a single copy gene and functions in the regulation of cell proliferation. Mutation of p53 is linked with tumour development, and this may involve abnormal functioning of mutant p53 protein. A mutant allele of p53 is functionally temperature-sensitive and can promote or suppress cell proliferation. The tertiary structure of the mutant protein is also sensitive to temperature and adopts promoter and suppressor forms of p53. A conformation model for the functioning of p53 proposes that wild-type p53 is induced to change from suppressor to promoter form during the cell growth response. This model predicts that any mutation that deregulates the normal control of p53 conformation may lead to cancer.