Mathematical model for chemotherapeutic drug efficacy in arresting tumour growth based on the cancer stem cell hypothesis

OBJECTIVE: Cancer stem cells have been identified as the growth root for various malignant tumours and are thought to be responsible for cancer recurrence following treatment. MATERIALS AND METHODS: Here, a predictive mathematical model for the cancer stem cell hypothesis is used to understand tumour responses to chemotherapeutic drugs and judge the efficacy of treatments in arresting tumour growth. The impact of varying drug efficacies on different abnormal cell populations is investigated through the kinetics associated with their decline in response to therapy. RESULTS AND CONCLUSIONS: The model predicts the clinically established 'dandelion phenomenon' and suggests that the best response to chemotherapy occurs when a drug targets abnormal stem cells. We compare continuous and periodic drug infusion. For the latter, we examine the relative importance of the drug cell-kill rate and the mean time between successive therapies, to identify the key attributes for successful treatment.     

The stem cell hypothesis in head and neck cancer

Cancer stem cells (CSCs) are tumoral cells which have stem features such as self-renewal, high migration capacity, drug resistance, high proliferation abilities. In the last 10 years the pathological meaning and the existence of CSCs have been matter of discussion and a large number of articles have been published about the role that these cells play in the development and maintenance of the tumors. Head and neck squamous-cell carcinoma (HNSCC) is the sixth most common cancer worldwide: early diagnosis of high-risk premalignant lesions are high priorities for reducing deaths due to head and neck cancer. In the last years the CSCs hypothesis has been faced also for head and neck cancer, with the aim of a better comprehension of the tumor biology and an early diagnosis. The evidence that the development of a tumor comes from a small number of cells with stem-like characteristic, could bring too to the identification of therapies against these cellular target, fundamental for maintenance and progression of the lesion. Here, a literature review has been reported about the detection of supposed CSCs in head and neck 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.     

Mathematical model for contact inhibited cell division

We present a mathematical model for cell growth, which takes into account cell-cell interactions and leads to non-exponential inhibited growth of number of cells. The resulting difference equation is solved and extended to a differential equation which turns out to be of a non-linear diffusion type.     

Three‐dimensional cell culture model utilization in cancer stem cell research

Three‐dimensional (3D) cell culture models are becoming increasingly popular in contemporary cancer research and drug resistance studies. Recently, scientists have begun incorporating cancer stem cells (CSCs) into 3D models and modifying culture components in order to mimic in vivo conditions better. Currently, the global cell culture market is primarily focused on either 3D cancer cell cultures or stem cell cultures, with less focus on CSCs. This is evident in the low product availability officially indicated for 3D CSC model research. This review discusses the currently available commercial products for CSC 3D culture model research. Additionally, we discuss different culture media and components that result in higher levels of stem cell subpopulations while better recreating the tumor microenvironment. In summary, although progress has been made applying 3D technology to CSC research, this technology could be further utilized and a greater number of 3D kits dedicated specifically to CSCs should be implemented.     

Extinction models for cancer stem cell therapy

Cells with stem cell-like properties are now viewed as initiating and sustaining many cancers. This suggests that cancer can be cured by driving these cancer stem cells to extinction. The problem with this strategy is that ordinary stem cells are apt to be killed in the process. This paper sets bounds on the killing differential (difference between death rates of cancer stem cells and normal stem cells) that must exist for the survival of an adequate number of normal stem cells. Our main tools are birth-death Markov chains in continuous time. In this framework, we investigate the extinction times of cancer stem cells and normal stem cells. Application of extreme value theory from mathematical statistics yields an accurate asymptotic distribution and corresponding moments for both extinction times. We compare these distributions for the two cell populations as a function of the killing rates. Perhaps a more telling comparison involves the number of normal stem cells N_H at the extinction time of the cancer stem cells. Conditioning on the asymptotic time to extinction of the cancer stem cells allows us to calculate the asymptotic mean and variance of N_H. The full distribution of N_H can be retrieved by the finite Fourier transform and, in some parameter regimes, by an eigenfunction expansion. Finally, we discuss the impact of quiescence (the resting state) on stem cell dynamics. Quiescence can act as a sanctuary for cancer stem cells and imperils the proposed therapy. We approach the complication of quiescence via multitype branching process models and stochastic simulation. Improvements to the τ-leaping method of stochastic simulation make it a versatile tool in this context. We conclude that the proposed therapy must target quiescent cancer stem cells as well as actively dividing cancer stem cells. The current cancer models demonstrate the virtue of attacking the same quantitative questions from a variety of modeling, mathematical, and computational perspectives.     

Mathematical model of red cell sickling

A mathematical model of red cell sickling without crisis was established in vitro as a function of the parameters on which it depends: the oxygen partial pressure, the abnormal hemoglobin S concentration, the temperature and the pH. This model was verified both for the homozygotic sickling SS and heterozygotic sickling SC. Such a model allows systematic investigation of patients. Moreover, it opens up the way for rational comparison in the testing of drugs which are effective in the treatment of red cell sickling.     

Human papillomavirus and tar hypothesis for squamous cell cervical cancer

Cervical cancer is the second most common life-threatening cancer among women worldwide, with incidence rates ranging from 4.8 per 100,000 in the Middle East to 44.3 per 100,000 in East Africa. Human papillomavirus (HPV) infection, especially HPV-16 and HPV-18, plays a major role in the etiology of cervical cancer, but HPV alone is not sufficient to induce cancer. We propose that squamous cell cervical cancer is caused by an interaction of oncogenic viruses and cervical tar exposures. Cervical tar exposures occur from cigarette smoking, use of tar-based vaginal douche products (TBD), and long years of inhaling smoke from wood- and coal-burning stoves in poorly ventilated kitchens.     

The cancer stem cell: the breast cancer driver

Recent research in a large variety of tumors, including breast cancer, has given support to the "cancer stem cell hypothesis". Based on this, tumors contain and are driven by a cellular subcomponent that retains key stem cell properties. These include self-renewal, which drives tumorigenesis, and the capacity to generate cellular heterogeneity. Recently, different techniques have been used to isolate potential breast cancer stem cells with the cell surface phenotype CD44+CD24-/low lin- or expressing Aldehyde dehydrogenase. This model has fundamental implications for breast cancer treatment. The development of specific therapeutics that target this population is an important focus for the future.     

Mathematical model of a cell size checkpoint

How cells regulate their size from one generation to the next has remained an enigma for decades. Recently, a molecular mechanism that links cell size and cell cycle was proposed in fission yeast. This mechanism involves changes in the spatial cellular distribution of two proteins, Pom1 and Cdr2, as the cell grows. Pom1 inhibits Cdr2 while Cdr2 promotes the G2 → M transition. Cdr2 is localized in the middle cell region (midcell) whereas the concentration of Pom1 is highest at the cell tips and declines towards the midcell. In short cells, Pom1 efficiently inhibits Cdr2. However, as cells grow, the Pom1 concentration at midcell decreases such that Cdr2 becomes activated at some critical size. In this study, the chemistry of Pom1 and Cdr2 was modeled using a deterministic reaction-diffusion-convection system interacting with a deterministic model describing microtubule dynamics. Simulations mimicked experimental data from wild-type (WT) fission yeast growing at normal and reduced rates; they also mimicked the behavior of a Pom1 overexpression mutant and WT yeast exposed to a microtubule depolymerizing drug. A mechanism linking cell size and cell cycle, involving the downstream action of Cdr2 on Wee1 phosphorylation, is proposed.     

Determining cancer risk: the evolutionary multistage model or total stem cell divisions?

A recent hypothesis proposed that the total number of stem cell divisions in a tissue (TSCD model) determine its intrinsic cancer risk; however, a different model—the multistage model—has long been used to understand how cancer originates. Identifying the correct model has important implications for interpreting the frequency of cancers. Using worldwide cancer incidence data, we applied three tests to the TSCD model and an evolutionary multistage model of carcinogenesis (EMMC), a model in which cancer suppression is recognized as an evolving trait, with natural selection acting to suppress cancers causing a significant mean loss of Darwinian fitness. Each test supported the EMMC but contradicted the TSCD model. This outcome undermines results based on the TSCD model quantifying the relative importance of ‘bad luck'' (the random accumulation of somatic mutations) versus environmental and genetic factors in determining cancer incidence. Our testing supported the EMMC prediction that cancers of large rapidly dividing tissues predominate late in life. Another important prediction is that an indicator of recent oncogenic environmental change is an unusually high mean fitness loss due to cancer, rather than a high lifetime incidence. The evolutionary model also predicts that large and/or long-lived animals have evolved mechanisms of cancer suppression that may be of value in preventing or controlling human cancers.     

Limitations of the cancer stem cell theory

Cancer stem cells (CSCs) can be operationally defined as a subset of neoplastic cells which are responsible for the growth and re-growth of primary and metastatic tumors. Although the existence of perpetually dividing cells is a logical necessity to explain the malignant properties of human tumors, experimental data supporting their existence have only recently been obtained. New knowledge in basic stem cell biology and the availability of several cell surface markers for the definition and isolation of small subsets of immature cells coupled to the use of the classical model of xenotransplantation in immune deficient mice has identified putative CSCs in several solid tumors such as mammary, colon, brain, pancreas, prostate, melanoma and others. However, the theory must be considered as still in its infancy, since tumors grown in mice only partially recapitulate the biology of human cells. In addition, whether the “transformed” cell is the neoplastic counterpart of a normal stem cell or whether complete malignant behaviour can occur in a more differentiated cell has still to be demonstrated. In spite of these difficulties, the CSC hypothesis could be of clinical relevance, especially in the definition of new ways to assay drug sensitivity of primary human tumors.     

Cancer stem cell hypothesis: a brief summary and two proposals

The investigation and development of the cancer stem cell (CSC) model has received much focus during these years. CSC is characterized as a small fraction of cancer cells that have an indefinite ability for self-renewal and pluripotency and are responsible for initiating and sustaining of the bulk of cancer. So, whether current treatment strategies, most of which target the rapid division of cancer cells, could interfere with the slow-cycling CSCs is broadly questioned. Meanwhile, however, the new understanding of tumorigenesis has led to the development of new drug screening strategies. Both stem cells and mesenchymal stem cells have been vigorously used in pre-clinical studies of their anti-tumor potential, mainly due to their inherent tumoritropic migratory properties and their ability to carry anti-tumor transgenes. Here, based on the tumorigenic and tumoritropic characteristics of CSCs, we proposed two hypotheses exploring possible usage of CSCs as novel anti-tumor agents and potential sources for tissue regeneration. Further experimental validation of these hypotheses may unravel some new research topics.     

Rare corneal clones in mice suggest an age-related decrease of stem cell activity and support the limbal epithelial stem cell hypothesis

The anterior ocular surface comprises the cornea, conjunctiva and a narrow intermediate region called the limbus. It is widely accepted that the corneal epithelium is maintained by stem cells but different hypotheses propose that the stem cells that maintain the mouse corneal epithelium during normal homeostasis are located either in the basal limbal epithelium or throughout the basal corneal epithelium. There are no specific markers to help test these alternatives and new methods are required to distinguish between them. We observed that KRT5^LacZ/− transgenic mice produced rare β-galactosidase (β-gal)-positive radial stripes in the corneal epithelium. These stripes are likely to be clonal lineages of cells derived from stem cells, so they provide a lineage marker for actively proliferating stem cells. The distributions of the β-gal-positive radial stripes suggested they extended centripetally from the limbus, supporting the limbal epithelial stem cell (LESC) hypothesis. Stripe frequency declined between 15 and 30 weeks, which predicts a reduction in stem cell function with age. Pax6^+/−, KRT5^LacZ/− corneas had small patches rather than stripes, which confirms that corneal maintenance is abnormal in Pax6^+/− mice.