Spatial invasion dynamics on random and unstructured meshes: implications for heterogeneous tumor populations.

In this work we discuss a spatial evolutionary model for a heterogeneous cancer cell population. We consider the gain-of-function mutations that not only change the fitness potential of the mutant phenotypes against normal background cells but may also increase the relative motility of the mutant cells. The spatial modeling is implemented as a stochastic evolutionary system on a structured grid (a lattice, with random neighborhoods, which is not necessarily bi-directional) or on a two-dimensional unstructured mesh, i.e. a bi-directional graph with random numbers of neighbors. We present a computational approach to investigate the fixation probability of mutants in these spatial models. Additionally, we examine the effect of the migration potential on the spatial dynamics of mutants on unstructured meshes. Our results suggest that the probability of fixation is negatively correlated with the width of the distribution of the neighborhood size. Also, the fixation probability increases given a migration potential for mutants. We find that the fixation probability (of advantaged, disadvantaged and neutral mutants) on unstructured meshes is relatively smaller than the corresponding results on regular grids. More importantly, in the case of neutral mutants the introduction of a migration potential has a critical effect on the fixation probability and increases this by orders of magnitude. Further, we examine the effect of boundaries and as intuitively expected, the fixation probability is smaller on the boundary of regular grids when compared to its value in the bulk. Based on these computational results, we speculate on possible better therapeutic strategies that may delay tumor progression to some extent.

Quantitative approaches to cancer stem cells and epithelial-mesenchymal transition.

The last decade has witnessed significant advances in the application of mathematical and computational models to biological systems, especially to cancer biology. Here, we present stochastic and deterministic models describing tumour growth based on the cancer stem cell hypothesis, and discuss the application of these models to the epithelial-mesenchymal transition. In particular, we discuss how such quantitative approaches can be used to validate different possible scenarios that can lead to an increase in stem cell activity following induction of epithelial-mesenchymal transition, observed in recent experimental studies on human breast cancer and related cell lines. The utility of comparing mammosphere data to computational mammosphere simulations in elucidating the growth characteristics of mammary (cancer) stem cells is discussed as well.

Investigating the link between epithelial-mesenchymal transition and the cancer stem cell phenotype: A mathematical approach.

Under the cancer stem cell (CSC) hypothesis, sustained metastatic growth requires the dissemination of a CSC from the primary tumour followed by its re-establishment in a secondary site. The epithelial-mesenchymal transition (EMT), a differentiation process crucial to normal development, has been implicated in conferring metastatic ability on carcinomas. Balancing these two concepts has led researchers to investigate a possible link between EMT and the CSC phenotype-indeed, recent evidence indicates that, following induction of EMT in human breast cancer and related cell lines, stem cell activity increased, as judged by the presence of cells displaying the CD44(high)/CD24(low) phenotype and an increase in the ability of cells to form mammospheres. We mathematically investigate the nature of this increase in stem cell activity. A stochastic model is used when small number of cells are under consideration, namely in simulating the mammosphere assay, while a related continuous model is used to probe the dynamics of larger cell populations. Two scenarios of EMT-mediated CSC enrichment are considered. In the first, differentiated cells re-acquire a CSC phenotype-this model implicates fully mature cells as key subjects of de-differentiation and entails a delay period of several days before de-differentiation occurs. In the second, pre-existing CSCs experience accelerated division and increased proportion of self-renewing divisions; a lack of perfect CSC biomarkers and cell sorting techniques requires that this model be considered, further emphasizing the need for better characterization of the mammary (cancer) stem cell hierarchy. Additionally, we suggest the utility of comparing mammosphere data to computational mammosphere simulations in elucidating the growth characteristics of mammary (cancer) stem cells.

Dynamics of tumor growth and combination of anti-angiogenic and cytotoxic therapies.

Tumors cannot grow beyond a certain size (about 1-2 mm in diameter) through simple diffusion of oxygen and other essential nutrients into the tumor. Angiogenesis, the formation of blood vessels from pre-existing vessels, is a crucial and observed step, through which a tumor obtains its own blood supply. Thus, strategies that interfere with the development of this tumor vasculature, known as anti-angiogenic therapy, represent a novel approach to controlling tumor growth. Several pre-clinical studies have suggested that currently available angiogenesis inhibitors are unlikely to yield significant sustained improvements in tumor control on their own, but rather will need to be used in combination with conventional treatments to achieve maximal benefit. Optimal sequencing of anti-angiogenic treatment and radiotherapy or chemotherapy is essential to the success of these combined treatment strategies. Hence, a major challenge to mathematical modeling and computer simulations is to find appropriate dosages, schedules and sequencing of combination therapies to control or eliminate tumor growth. Here, we present a mathematical model that incorporates tumor cells and the vascular network, as well as their interplay. We can then include the effects of two different treatments, conventional cytotoxic therapy and anti-angiogenic therapy. The results are compared with available experimental and clinical data.

Mathematical modeling of brain tumors: effects of radiotherapy and chemotherapy.

Gliomas, the most common primary brain tumors, are diffusive and highly invasive. The standard treatment for brain tumors consists of a combination of surgery, radiation therapy and chemotherapy. Over the past few years, mathematical models have been applied to study untreated and treated brain tumors. In an effort to improve treatment strategies, we consider a simple spatio-temporal mathematical model, based on proliferation and diffusion, that incorporates the effects of radiotherapeutic and chemotherapeutic treatments. We study the effects of different schedules of radiation therapy, including fractionated and hyperfractionated external beam radiotherapy, using a generalized linear quadratic (LQ) model. The results are compared with published clinical data. We also discuss the results for combination therapy (radiotherapy plus temozolomide, a new chemotherapy agent), as proposed in recent clinical trials. We use the model to predict optimal sequencing of the postoperative (combination of radiotherapy and adjuvant, neo-adjuvant or concurrent chemotherapy) treatments for brain tumors.

Genotype by random environmental interactions gives an advantage to non-favored minor alleles.

Fixation probability, the probability that the frequency of a newly arising mutation in a population will eventually reach unity, is a fundamental quantity in evolutionary genetics. Here we use a number of models (several versions of the Moran model and the haploid Wright-Fisher model) to examine fixation probabilities for a constant size population where the fitness is a random function of both allelic state and spatial position, despite neither allele being favored on average. The concept of fitness varying with respect to both genotype and environment is important in models of cancer initiation and progression, bacterial dynamics, and drug resistance. Under our model spatial heterogeneity redefines the notion of neutrality for a newly arising mutation, as such mutations fix at a higher rate than that predicted under neutrality. The increased fixation probability appears to be due to rare alleles having an advantage. The magnitude of this effect can be large, and is an increasing function of the spatial variance and skew in fitness. The effect is largest when the fitness values of the mutants and wild types are anti-correlated across environments. We discuss results for both a spatial ring geometry of cells (such as that of a colonic crypt), a 2D lattice and a mass-action (complete graph) arrangement.

The constitutive properties of the brain paraenchyma Part 2. Fractional derivative approach.

Fractional models have proven to be very useful for studying viscoelastic materials. We consider the fractional Zener model (also called four-parameter model) to study both the relaxation function and creep compliance. The analytical results are compared with the known experimental results of the human brain tissue to obtain the best fit and brain mechanical parameters. The results are also compared to the non-fractional Zener model and four-parameter Burgers model, indicating that the four-parameter fractional model gives a substantially better fit for the all experimental data.

The constitutive properties of the brain parenchyma Part 1. Strain energy approach.

In this paper we study several constitutive equations for the brain based on the strain energy density function. We use the polynomial function and hyper-elastic Ogden model for the strain energy and include the energy dissipation by a Prony series expansion. The models are compared with known unconfined compression experimental results of the human brain tissue to obtain the best fitted model and brain mechanical parameters. Finite element simulations are also performed using the given constitutive equations, and numerical solutions match the analytical results very closely. The results are compared with other analytical and numerical calculations.

Frequency dependence of complex moduli of brain tissue using a fractional Zener model.

Brain tissue exhibits viscoelastic behaviour. If loading times are substantially short, static tests are not sufficient to determine the complete viscoelastic behaviour of the material, and dynamic test methods are more appropriate. The concept of complex modulus of elasticity is a powerful tool for characterizing the frequency domain behaviour of viscoelastic materials. On the other hand, it is well known that classical viscoelastic models can be generalized by means of fractional calculus to describe more complex viscoelastic behaviour of materials. In this paper, the fractional Zener model is investigated in order to describe the dynamic behaviour of brain tissue. The model is fitted to experimental data of oscillatory shear tests of bovine brain tissue to verify its behaviour and to obtain the material parameters.

Modeling Invasion Dynamics with Spatial Random-Fitness Due to Micro-Environment.

Numerous experimental studies have demonstrated that the microenvironment is a key regulator influencing the proliferative and migrative potentials of species. Spatial and temporal disturbances lead to adverse and hazardous microenvironments for cellular systems that is reflected in the phenotypic heterogeneity within the system. In this paper, we study the effect of microenvironment on the invasive capability of species, or mutants, on structured grids (in particular, square lattices) under the influence of site-dependent random proliferation in addition to a migration potential. We discuss both continuous and discrete fitness distributions. Our results suggest that the invasion probability is negatively correlated with the variance of fitness distribution of mutants (for both advantageous and neutral mutants) in the absence of migration of both types of cells. A similar behaviour is observed even in the presence of a random fitness distribution of host cells in the system with neutral fitness rate. In the case of a bimodal distribution, we observe zero invasion probability until the system reaches a (specific) proportion of advantageous phenotypes. Also, we find that the migrative potential amplifies the invasion probability as the variance of fitness of mutants increases in the system, which is the exact opposite in the absence of migration. Our computational framework captures the harsh microenvironmental conditions through quenched random fitness distributions and migration of cells, and our analysis shows that they play an important role in the invasion dynamics of several biological systems such as bacterial micro-habitats, epithelial dysplasia, and metastasis. We believe that our results may lead to more experimental studies, which can in turn provide further insights into the role and impact of heterogeneous environments on invasion dynamics.

Efficacy of dose escalation on TCP, recurrence and second cancer risks: a mathematical study.

OBJECTIVE: We investigated the effects of conventional and hypofractionation protocols by modelling tumour control probability (TCP) and tumour recurrence time, and examined their impact on second cancer risks. The main objectives of this study include the following: (a) incorporate tumour recurrence time and second cancer risks into the TCP framework and analyse the effects of variable doses and (b) investigate an efficient protocol to reduce the risk of a secondary malignancy while maximizing disease-free survival and tumour control. METHODS: A generalized mathematical formalism was developed that incorporated recurrence and second cancer risk models into the TCP dynamics. RESULTS: Our results suggest that TCP and relapse time are almost identical for conventional and hypofractionated regimens; however, second cancer risks resulting from hypofractionation were reduced by 22% when compared with the second cancer risk associated with a conventional protocol. The hypofractionated regimen appears to be sensitive to dose escalation and the corresponding impact on tumour recurrence time and reduction in second cancer risks. The reduction in second cancer risks is approximately 20% when the dose is increased from 60 to 72 Gy in a hypofractionated protocol. CONCLUSION: Our results suggest that hypofractionation may be a more efficient regimen in the context of TCP, relapse time and second cancer risks. Overall, our study demonstrates the importance of including a second cancer risk model in designing an efficient radiation regimen. ADVANCES IN KNOWLEDGE: The impact of various fractionation protocols on TCP and relapse in conjunction with second cancer risks is an important clinical question that is as yet unexplored.

Quantitative model for efficient temporal targeting of tumor cells and neovasculature.

The combination of cytotoxic therapies and antiangiogenic agents is emerging as a most promising strategy in the treatment of malignant tumors. However, the timing and sequencing of these treatments seem to play essential roles in achieving a synergic outcome. Using a mathematical modeling approach that is grounded on available experimental data, we investigate the spatial and temporal targeting of tumor cells and neovasculature with a nanoscale delivery system. Our model suggests that the experimental success of the nanoscale delivery system depends crucially on the trapping of chemotherapeutic agents within the tumor tissue. The numerical results also indicate that substantial further improvements in the efficiency of the nanoscale delivery system can be achieved through an adjustment of the temporal targeting mechanism.

Characterization of brain cancer stem cells: a mathematical approach.

OBJECTIVE: In recent years, support has increased for the notion that a subpopulation of brain tumour cells in possession of properties typically characteristic of stem cells is responsible for initiating and maintaining the tumour. Unravelling details of the brain tumour stem cell (BTSC) hierarchy, as well as interactions of these cells with various therapies, will be essential in the design of optimal treatment strategies. MATERIALS AND METHODS: Motivated by this, we have developed a mathematical model of the BTSC hypothesis that may aid in characterization of brain tumours, as well as in prediction of effective therapeutic strategies, which can be further validated in experimental and clinical studies. At the level of a small number of cells, the model developed herein is stochastic. For larger populations of cancer cells, the model is handled from a deterministic approach. RESULTS AND CONCLUSIONS: In the stochastic regime, importance of a relationship between the likelihoods of two distinct types of symmetric BTSC divisions in determining BTSC survival rates becomes apparent, consequently emphasizing the need for a set of biomarkers that are able to better characterize the BTSC hierarchy. At the large scale, we predict the importance of the aforementioned symmetric division rates in dictating brain tumour composition. Furthermore, we demonstrate possible therapeutic benefits of considering combination treatments of radiotherapy and putative BTSC inhibitors, such as bone morphogenetic proteins, while reinforcing the importance of developing novel treatment strategies that specifically target the BTSC subpopulation.

Mathematical modeling of ovarian cancer treatments: sequencing of surgery and chemotherapy.

Ovarian cancer has long been one of the most common forms of cancer in women. The main treatment for ovarian cancer comprises a combination of surgery and chemotherapy. In an effort to improve treatment strategies, a variety of mathematical models have been developed in the literature. In this paper, we consider a simple mathematical model that incorporates tumor growth as well as the effects of chemotherapeutic and surgical treatments in ovarian cancer. We consider several growth models and combine them with different cell-kill hypotheses. Surgery is assumed to eliminate a fixed fraction of tumor cells instantaneously. We discuss how different models predict the optimal sequencing of chemotherapeutic and surgical treatments. This work has been carried out in the context of ovarian cancer; however, the results may also be useful for other kind of cancers.

Hard-core yukawa model for charge-stabilized colloids

The hypernetted chain approximation is used to study the phase diagram of a simple hardcore Yukawa model of a charge-stabilized colloids. We calculate the static structure factor, the pair distribution function, and the collective mode energies over a wide range of parameters, and the results are used for studying the freezing transition of the system. The resulting phase diagram is in good agreement with the known estimates and the Monte Carlo simulations.