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.     

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.     

Exploring cancer stem cell niche directed tumor growth

The finding that only a sub-fraction of tumor cells, so called Cancer Stem Cells (CSC), is endowed with the capacity to initiate new tumors has important consequences for fundamental as well as clinical cancer research. Previously we established by computational modeling techniques that CSC driven tumor growth instigates infiltrative behavior, and perhaps most interesting, stimulates tumor cell heterogeneity. An important question that remains is to what extend CSC functions are intrinsically regulated or whether this capacity is orchestrated by the microenvironment, i.e. a putative CSC niche. Here we investigate how extrinsic regulation of CSC properties affects the characteristics of malignancies. We find that highly invasive growth in tumors dependent on a small subset of cells is not restricted to CSC-driven tumors, but is also observed in tumors where the CSC capacity of tumor cells is completely defined by the microenvironment. Importantly, also the high level of heterogeneity that was observed for CSC-driven tumors is preserved and partially even increased in malignancies with a microenvironmentally orchestrated CSC population. This indicates that invasive growth and high heterogeneity are fundamental properties of tumors fueled by a small population of tumor cells.     

Implications of a simple mathematical model to cancer cell population dynamics

Recent research in cancer progression and treatment indicates that many forms of cancer arise from the development of a small subpopulation of abnormal cancer stem cells (CSCs) that promote cancer growth and spread. Many potential treatments preferentially interact with cells at certain stages of the cell cycle by either selective killing or halting the cell cycle, such as intense, nanosecond-duration pulsed electric fields (nsPEFs). Simple mathematical models of unfed cancer cell populations at the plateau of their growth characteristics may estimate the long-term consequences of these treatments on proliferating and quiescent cell populations. Applying such a model with no transition from the quiescent to proliferating state shows that it is possible for the proliferating cell population to fall below 1 if the quiescent cell population obtains a sufficient competitive advantage with respect to nutrient consumption and/or survival rate. Introducing small, realistic transition rates did not appreciably alter short-term or long-term population behaviour, indicating that the predicted small cell population behaviour (< 1 cell) is not an artefact of the simpler model. Experimental observations of nsPEF-induced effects on the cell cycle suggest that such a model may serve as a first step in assessing the viability of a given cancer treatment in vitro prior to clinical application.     

A mathematical model for cell cycle-specific cancer virotherapy

Oncolytic viruses preferentially infect and replicate in cancerous cells, leading to elimination of tumour populations, while sparing most healthy cells. Here, we study the cell cycle-specific activity of viruses such as vesicular stomatitis virus (VSV). In spite of its capacity as a robust cytolytic agent, VSV cannot effectively attack certain tumour cell types during the quiescent, or resting, phase of the cell cycle. In an effort to understand the interplay between the time course of the cell cycle and the specificity of VSV, we develop a mathematical model for cycle-specific virus therapeutics. We incorporate the minimum biologically required time spent in the non-quiescent cell cycle phases using systems of differential equations with incorporated time delays. Through analysis and simulation of the model, we describe how varying the minimum cycling time and the parameters that govern viral dynamics affect the stability of the cancer-free equilibrium, which represents therapeutic success.     

A mathematical model of in vitro cancer cell growth and treatment with the antimitotic agent curacin A

A mathematical model of cancer cell growth and response to treatment with the experimental antimitotic agent curacin A is presented. Rate parameters for the untreated growth of MCF-7/LY2 breast cancer and A2780 ovarian cell lines are determined from in vitro growth studies. Subsequent growth studies following treatments with 2.5, 25 and 50 nanomolar (nM), concentrations of curacin A are used to determine effects on the cell cycle and cell viability. The model's system of ordinary differential equations yields an approximate analytical solution which predicts the minimum concentration necessary to prevent growth. The model shows that cell growth is arrested when the apoptotic rate is greater than the mitotic rate and that the S-phase transition rate acts to amplify this effect. Analysis of the data suggests that curacin A is rapidly absorbed into both cell lines causing an increase in the S-phase transition and a decrease in the M-phase transition. The model also indicates that the rate of apoptosis remains virtually constant for MCF-7/LY2 while that of A2780 increases 38% at 2.5 nM and 59% at 50 nM as compared to the untreated apoptotic rate.     

A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

A simple mathematical model for Ebola in Africa

We deal with the following question: Can the consumption of contaminated bush meat, the funeral practices and the environmental contamination explain the recurrence and persistence of Ebola virus disease outbreaks in Africa? We develop an SIR-type model which, incorporates both the direct and indirect transmissions in such a manner that there is a provision of Ebola viruses. We prove that the full model has one (endemic) equilibrium which is locally asymptotically stable whereas, it is globally asymptotically stable in the absence of the Ebola virus shedding in the environment. For the sub-model without the provision of Ebola viruses, the disease dies out or stabilizes globally at an endemic equilibrium. At the endemic level, the number of infectious is larger for the full model than for the sub-model without provision of Ebola viruses. We design a nonstandard finite difference scheme, which preserves the dynamics of the model. Numerical simulations are provided.     

Evidence for discrete cell kinetic subpopulations in mouse epidermis based on mathematical analysis

Abstract. Continuous (repeated) labelling studies in mouse epidermis indicate that nearly all cells are labelled after about 100 hr. Percentage labelled mitoses studies ([³H]TdR at 15.00 and 03.00 hours) have a first peak that does not reach 100% and has a half-width of about 10 hr. Small second and third peaks can be detected at about 90 and 180 hr, respectively. The changes with time in the number of labelled cells show a difference dependent on the time of day of [³H]TdR administration. Both curves show an early doubling in labelled cells which then decline, forming a peak of labelled cells. A second peak occurs at about 120 hr. This is followed by a progressive decline with no further peaks until values of about 1% labelling are obtained at 340 hr.
These experiments have been investigated mathematically. A computer programme has been devized that permits all three types of experiments to be analysed simultaneously. More importantly, it can analyse situations with a heterogeneity in cell cycle parameters in all proliferative subpopulations.
Various models for epidermal cell replacement have been considered. The data as a whole can best be explained if the basal layer contains at least two distinct subpopulations of cells and an exponentially decaying post-mitotic population with a half-life of about 30 hr. The proliferative sub-populations must be characterized by near integer differences in the length of cycle, the precursor (stem) compartment having the longer cycle. An inverse relationship is required for the length of S, i.e. the shortest time for the stem cells.
A full range of cell kinetic parameters can be calculated and are tabulated for the most appropriate model system which is one involving three transit proliferating subpopulations.     

A mathematical model of epiphyseal development: hypothesis of growth pattern of the secondary ossification centre

This paper introduces a 'hypothesis about the growth pattern of the secondary ossification centre (SOC)', whereby two phases are assumed. First, the formation of cartilage canals as an event essential for the development of the SOC. Second, once the canals are merged in the central zone of the epiphysis, molecular factors are released (primarily Runx2 and MMP9) spreading and causing hypertrophy of adjacent cells. In addition, there are two important molecular factors in the epiphysis: PTHrP and Ihh. The first one inhibits chondrocyte hypertrophy and the second helps the cell proliferation. Between these factors, there is negative feedback, which generates a highly localised and stable pattern over time. From a mathematical point of view, this pattern is similar to the patterns of Turing. The spread of Runx2 hypertrophies the cells from the centre to the periphery of the epiphysis until found with high levels of PTHrP to inhibit hypertrophy. This mechanism produces the epiphyseal bone-plate. Moreover, the hypertrophy is inhibited when the cells sense low shear stress and high pressure levels that maintain the articular cartilage structure. To test this hypothesis, we solve a system of coupled partial differential equations using the finite element method and we have obtained spatio-temporal patterns of the growth process of the SOC. The model is in qualitative agreement with experimental results previously reported by other authors. Thus, we conclude that this model can be used as a methodological basis to present a complete mathematical model of the whole epiphyseal development.     

A mathematical model of tumor growth and its response to single irradiation

Background

Mathematical modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying mathematical models of tumor response to radiation therapy. We developed a simple mathematical model to simulate the growth of tumor volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness.

Methods

We made several key assumptions of the model. Tumor volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. Tumor growth rate (or tumor volume doubling time) is proportional to the ratio of the volumes of tumor vasculature and the tumor. The vascular volume grows slower than the tumor by introducing the vascular growth retardation factor, θ. Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The model was applied to simulate pre-treatment growth and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS).

Results

By selecting appropriate model parameters, we showed the temporal variation of the tumors for both the rat experiment and the clinical GKSRS cases could be easily replicated by the simple model. Additionally, the application of our model to the GKSRS cases showed that the α-value, which is an indicator of radiation sensitivity in the LQ model, and the value of θ could be predictors of the post-treatment volume change.

Conclusions

The proposed model was successful in representing both the animal experimental data and the clinically observed tumor volume changes. We showed that the model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. However, there is a large statistical uncertainty of the result due to the small sample size. Therefore, a future clinical study with a larger number of patients is needed to confirm the finding.

     

A simple, leaky cell growth model for plant cell aggregates

A simple growth model is proposed for plant cell aggregates which accounts for leakage of a single intermediate metabolite from the aggregates to the medium. This model predicts a lag phase in the growth curve whose extent is determined by the intermediate metabolite leakage coefficient and its equilibrium distribution coefficient between the medium and the cell aggregates, the size of the inoculum relative to the system total water content, and the initial intermediate metabolite content in the medium. The model thus provides for an interaction between growing plant cells and their environment in a way that has heretofore been unquantified. Preliminary validation of the model has been made against literature data of Dioscorea deltoidea grown in batch suspension cell culture on sucrose, yielding a correlation coefficient of 0.997. The predicted glucose + fructose concentration in the medium agrees reasonably well with experimental measurements after ca, 3.5 days of culture, although a discrepancy exists between model prediction and experiment immediately after startup. Further validation of the model is suggested on this and other plant species.     

A simple kinetic model describes the processivity of myosin-v

Myosin-V is a motor protein responsible for organelle and vesicle transport in cells. Recent single-molecule experiments have shown that it is an efficient processive motor that walks along actin filaments taking steps of mean size close to 36 nm. A theoretical study of myosin-V motility is presented following an approach used successfully to analyze the dynamics of conventional kinesin but also taking some account of step-size variations. Much of the present experimental data for myosin-V can be well described by a two-state chemical kinetic model with three load-dependent rates. In addition, the analysis predicts the variation of the mean velocity and of the randomness-a quantitative measure of the stochastic deviations from uniform, constant-speed motion-with ATP concentration under both resisting and assisting loads, and indicates a substep of size d(0) approximately 13-14 nm (from the ATP-binding state) that appears to accord with independent observations.     

A simple mathematical modeling of the growth of microorganism colonies

The growth rates of aspergillus, fusarium, and penicillium microorganism colonies in Czapex Dox Agar as a feed material, under room conditions, are observed to be linear. This phenomenon is mathematically modeled and exactly predicted on the basis of the exponential growth assumption of a single microorganism. The approach allows an easy determination of the multiplication constant of a had microorganisms been allowed to grow freely microorganism, in given conditions.     

A simple kinetic model for myeloma cell culture with consideration of lysine limitation

A simple kinetic model is developed to describe the dynamic behavior of myeloma cell growth and cell metabolism. Glucose, glutamine as well as lysine are considered as growth limiting substrates. The cell growth was restricted as soon as the extracellular lysine is exhausted and then intracellular lysine becomes a growth limiting substrate. In addition, a metabolic regulator model together with the Monod model is used to deal with the growth lag phase after inoculation or feeding. By using these models, concentrations of substrates and metabolites, as well as densities of viable and dead cells are quantitatively described. One batch cultivation and two fed-batch cultivations with pulse feeding of nutrients are used to validate the model.     

Extracellular ATP reduces tumor sphere growth and cancer stem cell population in glioblastoma cells

Glioblastoma is the most aggressive tumor in the CNS and is characterized by having a cancer stem cell (CSC) subpopulation essential for tumor survival. The purinergic system plays an important role in glioma growth, since adenosine triphosphate (ATP) can induce proliferation of glioma cells, and alteration in extracellular ATP degradation by the use of exogenous nucleotidases dramatically alters the size of gliomas in rats. The aim of this work was to characterize the effect of the purinergic system on glioma CSCs. Human U87 glioma cultures presented tumor spheres that express the markers of glioma cancer stem cells CD133, Oct-4, and Nanog. Messenger RNA of several purinergic receptors were differently expressed in spheres when compared to a cell monolayer not containing spheres. Treatment of human gliomas U87 or U343 as well as rat C6 gliomas with 100 μM of ATP reduced the number of tumor spheres when grown in neural stem cell medium supplemented with epidermal growth factor and basic fibroblast growth factor. Moreover, ATP caused a decline in the number of spheres observed in culture in a dose-dependent manner. ATP also reduces the expression of Nanog, as determined by flow cytometry, as well as CD133 and Oct-4, as analyzed by flow cytometry and RT-PCR in U87 cells. The differential expression of purinergic receptor in tumor spheres when compared to adherent cells and the effect of ATP in reducing tumor spheres suggest that the purinergic system affects CSC biology and that ATP may be a potential agonist for differentiation therapy.     

A mathematical model for toxin accumulation by killer yeasts based on the yeast population growth

The accumulation of toxin by killer yeast populations is modelled starting from a mechanistic approach that explains the toxin production in terms of yeast population growth, and takes into account the environmental inactivation of the toxin. A modified Richard's general equation for limited growth is used to define the function that describes the toxin produced in relation to the yeast biomass increase. The relationship between the rates of cell and toxin production is explicitly shown, and the implications of the resulting proportionality factor are discussed. The model parameters have been adjusted and the model has been validated using experimental data of growth and toxin accumulation from cultures of Pichia membranaefaciens in two different media. The differences between both types of cultures are analysed on the basis of parameter estimates and the predicted rate of toxin production per cell. The results support the hypothesis that biomass production and toxin synthesis are controlled in different ways; they also suggest that the composition of the medium could have a distinct effect on toxin synthesis. Model assumptions are discussed in comparison with a previous model for killer-sensitive interaction of Saccharomyces cerevisiae strains.