A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells

A simple agent-based hybrid model to simulate the biophysics of glioblastoma multiforme cells and the concomitant evolution of the oxygen field

BACKGROUND AND OBJECTIVES

Glioblastoma multiforme (GBM) is one of the most aggressive cancers of the central nervous system. It is characterized by a high mitotic activity and an infiltrative ability of the glioma cells, neovascularization and necrosis. GBM evolution entails the continuous interplay between heterogeneous cell populations, chemotaxis, and physical cues through different scales. In this work, an agent-based hybrid model is proposed to simulate the coupling of the multiscale biological events involved in the GBM invasion, specifically the individual and collective migration of GBM cells and the concurrent evolution of the oxygen field and phenotypic plasticity. An asset of the formulation is that it is conceptually and computationally simple but allows to reproduce the complexity and the progression of the GBM micro-environment at cell and tissue scales simultaneously.

METHODS

The migration is reproduced as the result of the interaction between every single cell and its micro-environment. The behavior of each individual cell is formulated through genotypic variables whereas the cell micro-environment is modeled in terms of the oxygen concentration and the cell density surrounding each cell. The collective behavior is formulated at a cellular scale through a flocking model. The phenotypic plasticity of the cells is induced by the micro-environment conditions, considering five phenotypes.

RESULTS

The model has been contrasted by benchmark problems and experimental tests showing the ability to reproduce different scenarios of glioma cell migration. In all cases, the individual and collective cell migration and the coupled evolution of both the oxygen field and phenotypic plasticity have been properly simulated. This simple formulation allows to mimic the formation of relevant hallmarks of glioblastoma multiforme, such as the necrotic cores, and to reproduce experimental evidences related to the mitotic activity in pseudopalisades.

CONCLUSIONS

In the collective migration, the survival of the clusters prevails at the expense of cell mitosis, regardless of the size of the groups, which delays the formation of necrotic foci and reduces the rate of oxygen consumption.

On the role of mechanical signals on sprouting angiogenesis through computer modeling approaches

Sprouting angiogenesis, the formation of new vessels from preexisting vasculature, is an essential process in the regeneration of new tissues as well as in the development of some diseases like cancer. Although early studies identified chemical signaling as the main driver of this process, many recent studies have shown a strong role of mechanical signals in the formation of new capillaries. Different types of mechanical signals (e.g., external forces, cell traction forces, and blood flow-induced shear forces) have been shown to play distinct roles in the process; however, their interplay remains still largely unknown. During the last decades, mathematical and computational modeling approaches have been developed to investigate and better understand the mechanisms behind mechanically driven angiogenesis. In this manuscript, we review computational models of angiogenesis with a focus on models investigating the role of mechanics on the process. Our aim is not to provide a detailed review on model methodology but to describe what we have learnt from these models. We classify models according to the mechanical signals being investigated and describe how models have looked into their role on the angiogenic process. We show that a better understanding of the mechanobiology of the angiogenic process will require the development of computer models that incorporate the interactions between the multiple mechanical signals and their effect on cellular responses, since they all seem to play a key in sprout patterning. In the end, we describe some of the remaining challenges of computational modeling of angiogenesis and discuss potential avenues for future research.

Computational simulation of cellular proliferation using a meshless method

BACKGROUND AND OBJECTIVE

During cell proliferation, cells grow and divide in order to obtain two new genetically identical cells. Understanding this process is crucial to comprehend other biological processes. Computational models and algorithms have emerged to study this process and several examples can be found in the literature. The objective of this work was to develop a new computational model capable of simulating cell proliferation. This model was developed using the Radial Point Interpolation Method, a meshless method that, to the knowledge of the authors, was never used to solve this type of problem. Since the efficiency of the model strongly depends on the efficiency of the meshless method itself, the optimal numbers of integration points per integration cell and of nodes for each influence-domain were investigated. Irregular nodal meshes were also used to study their influence on the algorithm.

METHODS

For the first time, an iterative discrete model solved by the Radial Point Interpolation Method based on the Galerkin weak form was used to establish the system of equations from the reaction-diffusion integro-differential equations, following a new phenomenological law proposed by the authors that describes the growth of a cell over time while dependant on oxygen and glucose availability. The discretization flexibility of the meshless method allows to explicitly follow the geometric changes of the cell until the division phase.

RESULTS

It was found that an integration scheme of 6 × 6 per integration cell and influence-domains with only seven nodes allows to predict the cellular growth and division with the best balance between the relative error and the computing cost. Also, it was observed that using irregular meshes do not influence the solution.

CONCLUSIONS

Even in a preliminary phase, the obtained results are promising, indicating that the algorithm might be a potential tool to study cell proliferation since it can predict cellular growth and division. Moreover, the Radial Point Interpolation Method seems to be a suitable method to study this type of process, even when irregular meshes are used. However, to optimize the algorithm, the integration scheme and the number of nodes inside the influence-domains must be considered.

DNA-structured mathematical model of cell-cycle progression in cyclic hypoxia

New experimental data have shown how the periodic exposure of cells to low oxygen levels (i.e., cyclic hypoxia) impacts their progress through the cell-cycle. Cyclic hypoxia has been detected in tumours and linked to poor prognosis and treatment failure. While fluctuating oxygen environments can be reproduced in vitro, the range of oxygen cycles that can be tested is limited. By contrast, mathematical models can be used to predict the response to a wide range of cyclic dynamics. Accordingly, in this paper we develop a mechanistic model of the cell-cycle that can be combined with in vitro experiments, to better understand the link between cyclic hypoxia and cell-cycle dysregulation. A distinguishing feature of our model is the inclusion of impaired DNA synthesis and cell-cycle arrest due to periodic exposure to severely low oxygen levels. Our model decomposes the cell population into five compartments and a time-dependent delay accounts for the variability in the duration of the S phase which increases in severe hypoxia due to reduced rates of DNA synthesis. We calibrate our model against experimental data and show that it recapitulates the observed cell-cycle dynamics. We use the calibrated model to investigate the response of cells to oxygen cycles not yet tested experimentally. When the re-oxygenation phase is sufficiently long, our model predicts that cyclic hypoxia simply slows cell proliferation since cells spend more time in the S phase. On the contrary, cycles with short periods of re-oxygenation are predicted to lead to inhibition of proliferation, with cells arresting from the cell-cycle in the G2 phase. While model predictions on short time scales (about a day) are fairly accurate (i.e, confidence intervals are small), the predictions become more uncertain over longer periods. Hence, we use our model to inform experimental design that can lead to improved model parameter estimates and validate model predictions.

Mathematical model and computational scheme for multi-phase modeling of cellular population and microenvironmental dynamics in soft tissue

In this paper we introduce a system of partial differential equations that is capable of modeling a variety of dynamic processes in soft tissue cellular populations and their microenvironments. The model is designed to be general enough to simulate such processes as tissue regeneration, tumor growth, immune response, and many more. It also has built-in flexibility to include multiple chemical fields and/or sub-populations of cells, interstitial fluid and/or extracellular matrix. The model is derived from the conservation laws for mass and linear momentum and therefore can be classified as a continuum multi-phase model. A careful choice of state variables provides stability in solving the system of discretized equations defining advective flux terms. A concept of deviation from normal allows us to use simplified constitutive relations for stresses. We also present an algorithm for computing numerical approximations to the solutions of the system and discuss properties of these approximations. We demonstrate several examples of applications of the model. Numerical simulations show a significant potential of the model for simulating a variety of processes in soft tissues.

A persistent invasive phenotype in post-hypoxic tumor cells is revealed by fate mapping and computational modeling

Hypoxia is a critical factor in solid tumors that has been associated with cancer progression and aggressiveness. We recently developed a hypoxia fate mapping system to trace post-hypoxic cells within a tumor for the first time. This approach uses an oxygen-dependent fluorescent switch and allowed us to measure key biological features such as oxygen distribution, cell proliferation, and migration. We developed a computational model to investigate the motility and phenotypic persistence of hypoxic and post-hypoxic cells during tumor progression. The cellular behavior was defined by phenotypic persistence time, cell movement bias, and the fraction of cells that respond to an enhanced migratory stimulus. This work combined advanced cell tracking and imaging techniques with mathematical modeling, to reveal that a persistent invasive migratory phenotype that develops under hypoxia is required for cellular escape into the surrounding tissue, promoting the formation of invasive structures (“plumes”) that expand toward the oxygenated tumor regions.

•

A fluorescent fate mapping system allows tracking of hypoxic and post-hypoxic cells

•

Computational modeling predicts the formation of post-hypoxic invasive plumes

•

Simulations show post-hypoxic cells must maintain persistant migration to form plumes

•

Tracking cells exposed to intratumoral hypoxia confirms persistent migration

In silico simulation of the effect of hypoxia on MCF-7 cell cycle kinetics under fractionated radiotherapy

The treatment outcome of a given fractionated radiotherapy scheme is affected by oxygen tension and cell cycle kinetics of the tumor population. Numerous experimental studies have supported the variability of radiosensitivity with cell cycle phase. Oxygen modulates the radiosensitivity through hypoxia-inducible factor (HIF) stabilization and oxygen fixation hypothesis (OFH) mechanism. In this study, an existing mathematical model describing cell cycle kinetics was modified to include the oxygen-dependent G1/S transition rate and radiation inactivation rate. The radiation inactivation rate used was derived from the linear-quadratic (LQ) model with dependence on oxygen enhancement ratio (OER), while the oxygen-dependent correction for the G1/S phase transition was obtained from numerically solving the ODE system of cyclin D-HIF dynamics at different oxygen tensions. The corresponding cell cycle phase fractions of aerated MCF-7 tumor population, and the resulting growth curve obtained from numerically solving the developed mathematical model were found to be comparable to experimental data. Two breast radiotherapy fractionation schemes were investigated using the mathematical model. Results show that hypoxia causes the tumor to be more predominated by the tumor subpopulation in the G1 phase and decrease the fractional contribution of the more radioresistant tumor cells in the S phase. However, the advantage provided by hypoxia in terms of cell cycle phase distribution is largely offset by the radioresistance developed through OFH. The delayed proliferation caused by severe hypoxia slightly improves the radiotherapy efficacy compared to that with mild hypoxia for a high overall treatment duration as demonstrated in the 40-Gy fractionation scheme.

Targeting Cellular DNA Damage Responses in Cancer: An In Vitro-Calibrated Agent-Based Model Simulating Monolayer and Spheroid Treatment Responses to ATR-Inhibiting Drugs

We combine a systems pharmacology approach with an agent-based modelling approach to simulate LoVo cells subjected to AZD6738, an ATR (ataxia–telangiectasia-mutated and rad3-related kinase) inhibiting anti-cancer drug that can hinder tumour proliferation by targeting cellular DNA damage responses. The agent-based model used in this study is governed by a set of empirically observable rules. By adjusting only the rules when moving between monolayer and multi-cellular tumour spheroid simulations, whilst keeping the fundamental mathematical model and parameters intact, the agent-based model is first parameterised by monolayer in vitro data and is thereafter used to simulate treatment responses in in vitro tumour spheroids subjected to dynamic drug delivery. Spheroid simulations are subsequently compared to in vivo data from xenografts in mice. The spheroid simulations are able to capture the dynamics of in vivo tumour growth and regression for approximately 8 days post-tumour injection. Translating quantitative information between in vitro and in vivo research remains a scientifically and financially challenging step in preclinical drug development processes. However, well-developed in silico tools can be used to facilitate this in vitro to in vivo translation, and in this article, we exemplify how data-driven, agent-based models can be used to bridge the gap between in vitro and in vivo research. We further highlight how agent-based models, that are currently underutilised in pharmaceutical contexts, can be used in preclinical drug development.

A multi-scale agent-based model for avascular tumour growth

In this paper, we have developed a multi-scale, lattice-free, agent based model of avascular tumour growth in epithelial tissue. The model integrates different events to identify the underlying diversity within intracellular, cellular, and extracellular layer dynamics. The model considers every cell as an agent. A cellular agent may proliferate, spawns two identical daughter agents, or it may be transformed into other phenotypes during its life time depending on its internal proteins' activity as well as its external microenvironment. In this context, a simplified age-structured cell cycle model is adopted from the existing literature. The model considers that the intracellular events are regulated by p27 gene expression. In this model, p27 protein controls the overall tumour growth dynamics. Moreover, p27 is controlled by the external oxygen and nutrients that are modelled with the reaction-diffusion equations. The model also considers several biophysical forces which directly effect on the tumour growth dynamics. This modelling framework offers biologically realistic outcomes and also covers important criteria of the hallmarks of cancer which include oxygen and nutrient consumptions, micro-environmental heterogeneity, tumour cell proliferation by avoiding growth suppressor signals, replication of tumour cells at an abnormally faster rate, and resistance of apoptosis. The avascular tumour growth model is validated with immunohistochemistry and histopathology data. The outcome of the proposed model is very close to the range of the patient data, which concludes that the model is capable enough to mimic these complex biophysical phenomena.

Combining hypoxia-activated prodrugs and radiotherapy in silico: Impact of treatment scheduling and the intra-tumoural oxygen landscape

Hypoxia-activated prodrugs (HAPs) present a conceptually elegant approach to not only overcome, but better yet, exploit intra-tumoural hypoxia. Despite being successful in vitro and in vivo, HAPs are yet to achieve successful results in clinical settings. It has been hypothesised that this lack of clinical success can, in part, be explained by the insufficiently stringent clinical screening selection of determining which tumours are suitable for HAP treatments. Taking a mathematical modelling approach, we investigate how tumour properties and HAP-radiation scheduling influence treatment outcomes in simulated tumours. The following key results are demonstrated in silico: (i) HAP and ionising radiation (IR) monotherapies may attack tumours in dissimilar, and complementary, ways. (ii) HAP-IR scheduling may impact treatment efficacy. (iii) HAPs may function as IR treatment intensifiers. (iv) The spatio-temporal intra-tumoural oxygen landscape may impact HAP efficacy. Our in silico framework is based on an on-lattice, hybrid, multiscale cellular automaton spanning three spatial dimensions. The mathematical model for tumour spheroid growth is parameterised by multicellular tumour spheroid (MCTS) data.

Multiscale Agent-Based and Hybrid Modeling of the Tumor Immune Microenvironment

Multiscale systems biology and systems pharmacology are powerful methodologies that are playing increasingly important roles in understanding the fundamental mechanisms of biological phenomena and in clinical applications. In this review, we summarize the state of the art in the applications of agent-based models (ABM) and hybrid modeling to the tumor immune microenvironment and cancer immune response, including immunotherapy. Heterogeneity is a hallmark of cancer; tumor heterogeneity at the molecular, cellular, and tissue scales is a major determinant of metastasis, drug resistance, and low response rate to molecular targeted therapies and immunotherapies. Agent-based modeling is an effective methodology to obtain and understand quantitative characteristics of these processes and to propose clinical solutions aimed at overcoming the current obstacles in cancer treatment. We review models focusing on intra-tumor heterogeneity, particularly on interactions between cancer cells and stromal cells, including immune cells, the role of tumor-associated vasculature in the immune response, immune-related tumor mechanobiology, and cancer immunotherapy. We discuss the role of digital pathology in parameterizing and validating spatial computational models and potential applications to therapeutics.

Modeling triple-negative breast cancer heterogeneity: Effects of stromal macrophages, fibroblasts and tumor vasculature

A hallmark of breast tumors is its spatial heterogeneity that includes its distribution of cancer stem cells and progenitor cells, but also heterogeneity in the tumor microenvironment. In this study we focus on the contributions of stromal cells, specifically macrophages, fibroblasts, and endothelial cells on tumor progression. We develop a computational model of triple-negative breast cancer based on our previous work and expand it to include macrophage infiltration, fibroblasts, and angiogenesis. In vitro studies have shown that the secretomes of tumor-educated macrophages and fibroblasts increase both the migration and proliferation rates of triple-negative breast cancer cells. In vivo studies also demonstrated that blocking signaling of selected secreted factors inhibits tumor growth and metastasis in mouse xenograft models. We investigate the influences of increased migration and proliferation rates on tumor growth, the effect of the presence on fibroblasts or macrophages on growth and morphology, and the contributions of macrophage infiltration on tumor growth. We find that while the presence of macrophages increases overall tumor growth, the increase in macrophage infiltration does not substantially increase tumor growth and can even stifle tumor growth at excessive rates.

Mathematical modelling of interacting mechanisms for hypoxia mediated cell cycle commitment for mesenchymal stromal cells

Background

Existing experimental data have shown hypoxia to be an important factor affecting the proliferation of mesenchymal stromal cells (MSCs), but the contrasting observations made at various hypoxic levels raise the questions of whether hypoxia accelerates proliferation, and how. On the other hand, in order to meet the increasing demand of MSCs, an optimised bioreactor control strategy is needed to enhance in vitro production.

Results

A comprehensive, single-cell mathematical model has been constructed in this work, which combines cellular oxygen sensing with hypoxia-mediated cell cycle progression to predict cell cycle commitment as a proxy to proliferation rate. With oxygen levels defined for in vitro cell culture, the model predicts enhanced proliferation under intermediate (2–8%) and mild (8–15%) hypoxia and cell quiescence under severe (< 2%) hypoxia. Global sensitivity analysis and quasi-Monte Carlo simulation revealed that within a certain range (+/− 100%), model parameters affect (with varying significance) the minimum commitment time, but the existence of a range of optimal oxygen tension could be preserved with the hypothesized effects of Hif2α and reactive oxygen species (ROS). It appears that Hif2α counteracts Hif1α and ROS-mediated protein deactivation under intermediate hypoxia and normoxia (20%), respectively, to regulate the response of cell cycle commitment to oxygen tension.

Conclusion

Overall, this modelling study offered an integrative framework to capture several interacting mechanisms and allowed in silico analysis of their individual and collective roles in shaping the hypoxia-mediated commitment to cell cycle. The model offers a starting point to the establishment of a suitable mechanism that can satisfactorily explain the different existing experimental observations from different studies, and warrants future extension and dedicated experimental validation to eventually support bioreactor optimisation.

Electronic supplementary material

The online version of this article (10.1186/s12918-018-0560-3) contains supplementary material, which is available to authorized users.

Capturing the Dynamics of a Hybrid Multiscale Cancer Model with a Continuum Model

Cancer is a complex disease involving processes at spatial scales from subcellular, like cell signalling, to tissue scale, such as vascular network formation. A number of multiscale models have been developed to study the dynamics that emerge from the coupling between the intracellular, cellular and tissue scales. Here, we develop a continuum partial differential equation model to capture the dynamics of a particular multiscale model (a hybrid cellular automaton with discrete cells, diffusible factors and an explicit vascular network). The purpose is to test under which circumstances such a continuum model gives equivalent predictions to the original multiscale model, in the knowledge that the system details are known, and differences in model results can be explained in terms of model features (rather than unknown experimental confounding factors). The continuum model qualitatively replicates the dynamics from the multiscale model, with certain discrepancies observed owing to the differences in the modelling of certain processes. The continuum model admits travelling wave solutions for normal tissue growth and tumour invasion, with similar behaviour observed in the multiscale model. However, the continuum model enables us to analyse the spatially homogeneous steady states of the system, and hence to analyse these waves in more detail. We show that the tumour microenvironmental effects from the multiscale model mean that tumour invasion exhibits a so-called pushed wave when the carrying capacity for tumour cell proliferation is less than the total cell density at the tumour wave front. These pushed waves of tumour invasion propagate by triggering apoptosis of normal cells at the wave front. Otherwise, numerical evidence suggests that the wave speed can be predicted from linear analysis about the normal tissue steady state.

Coarse-graining and hybrid methods for efficient simulation of stochastic multi-scale models of tumour growth

The development of hybrid methodologies is of current interest in both multi-scale modelling and stochastic reaction-diffusion systems regarding their applications to biology. We formulate a hybrid method for stochastic multi-scale models of cells populations that extends the remit of existing hybrid methods for reaction-diffusion systems. Such method is developed for a stochastic multi-scale model of tumour growth, i.e. population-dynamical models which account for the effects of intrinsic noise affecting both the number of cells and the intracellular dynamics. In order to formulate this method, we develop a coarse-grained approximation for both the full stochastic model and its mean-field limit. Such approximation involves averaging out the age-structure (which accounts for the multi-scale nature of the model) by assuming that the age distribution of the population settles onto equilibrium very fast. We then couple the coarse-grained mean-field model to the full stochastic multi-scale model. By doing so, within the mean-field region, we are neglecting noise in both cell numbers (population) and their birth rates (structure). This implies that, in addition to the issues that arise in stochastic-reaction diffusion systems, we need to account for the age-structure of the population when attempting to couple both descriptions. We exploit our coarse-graining model so that, within the mean-field region, the age-distribution is in equilibrium and we know its explicit form. This allows us to couple both domains consistently, as upon transference of cells from the mean-field to the stochastic region, we sample the equilibrium age distribution. Furthermore, our method allows us to investigate the effects of intracellular noise, i.e. fluctuations of the birth rate, on collective properties such as travelling wave velocity. We show that the combination of population and birth-rate noise gives rise to large fluctuations of the birth rate in the region at the leading edge of front, which cannot be accounted for by the coarse-grained model. Such fluctuations have non-trivial effects on the wave velocity. Beyond the development of a new hybrid method, we thus conclude that birth-rate fluctuations are central to a quantitatively accurate description of invasive phenomena such as tumour growth.

Cell cycle time series gene expression data encoded as cyclic attractors in Hopfield systems

Modern time series gene expression and other omics data sets have enabled unprecedented resolution of the dynamics of cellular processes such as cell cycle and response to pharmaceutical compounds. In anticipation of the proliferation of time series data sets in the near future, we use the Hopfield model, a recurrent neural network based on spin glasses, to model the dynamics of cell cycle in HeLa (human cervical cancer) and S. cerevisiae cells. We study some of the rich dynamical properties of these cyclic Hopfield systems, including the ability of populations of simulated cells to recreate experimental expression data and the effects of noise on the dynamics. Next, we use a genetic algorithm to identify sets of genes which, when selectively inhibited by local external fields representing gene silencing compounds such as kinase inhibitors, disrupt the encoded cell cycle. We find, for example, that inhibiting the set of four kinases AURKB, NEK1, TTK, and WEE1 causes simulated HeLa cells to accumulate in the M phase. Finally, we suggest possible improvements and extensions to our model.

Cell cycle—the process in which a parent cell replicates its DNA and divides into two daughter cells—is an upregulated process in many forms of cancer. Identifying gene inhibition targets to regulate cell cycle is important to the development of effective therapies. Although modern high throughput techniques offer unprecedented resolution of the molecular details of biological processes like cell cycle, analyzing the vast quantities of the resulting experimental data and extracting actionable information remains a formidable task. Here, we create a dynamical model of the process of cell cycle using the Hopfield model (a type of recurrent neural network) and gene expression data from human cervical cancer cells and yeast cells. We find that the model recreates the oscillations observed in experimental data. Tuning the level of noise (representing the inherent randomness in gene expression and regulation) to the “edge of chaos” is crucial for the proper behavior of the system. We then use this model to identify potential gene targets for disrupting the process of cell cycle. This method could be applied to other time series data sets and used to predict the effects of untested targeted perturbations.

Systems Level Modeling of the Cell Cycle Using Budding Yeast

Proteins involved in the regulation of the cell cycle are highly conserved across all eukaryotes, and so a relatively simple eukaryote such as yeast can provide insight into a variety of cell cycle perturbations including those that occur in human cancer. To date, the budding yeast Saccharomyces cerevisiae has provided the largest amount of experimental and modeling data on the progression of the cell cycle, making it a logical choice for in-depth studies of this process. Moreover, the advent of methods for collection of high-throughput genome, transcriptome, and proteome data has provided a means to collect and precisely quantify simultaneous cell cycle gene transcript and protein levels, permitting modeling of the cell cycle on the systems level. With the appropriate mathematical framework and sufficient and accurate data on cell cycle components, it should be possible to create a model of the cell cycle that not only effectively describes its operation, but can also predict responses to perturbations such as variation in protein levels and responses to external stimuli including targeted inhibition by drugs. In this review, we summarize existing data on the yeast cell cycle, proteomics technologies for quantifying cell cycle proteins, and the mathematical frameworks that can integrate this data into representative and effective models. Systems level modeling of the cell cycle will require the integration of high-quality data with the appropriate mathematical framework, which can currently be attained through the combination of dynamic modeling based on proteomics data and using yeast as a model organism.

Simulation of head and neck cancer oxygenation and doubling time in a 4D cellular model with angiogenesis

Tumor oxygenation has been correlated with treatment outcome for radiotherapy. In this work, the dependence of tumor oxygenation on tumor vascularity and blood oxygenation was determined quantitatively in a 4D stochastic computational model of head and neck squamous cell carcinoma (HNSCC) tumor growth and angiogenesis. Additionally, the impacts of the tumor oxygenation and the cancer stem cell (CSC) symmetric division probability on the tumor volume doubling time and the proportion of CSCs in the tumor were also quantified. Clinically relevant vascularities and blood oxygenations for HNSCC yielded tumor oxygenations in agreement with clinical data for HNSCC. The doubling time varied by a factor of 3 from well oxygenated tumors to the most severely hypoxic tumors of HNSCC. To obtain the doubling times and CSC proportions clinically observed in HNSCC, the model predicts a CSC symmetric division probability of approximately 2% before treatment. To obtain the doubling times clinically observed during treatment when accelerated repopulation is occurring, the model predicts a CSC symmetric division probability of approximately 50%, which also results in CSC proportions of 30-35% during this time.

Development of an in silico stochastic 4D model of tumor growth with angiogenesis

PURPOSE

A stochastic computer model of tumour growth with spatial and temporal components that includes tumour angiogenesis was developed. In the current work it was used to simulate head and neck tumour growth. The model also provides the foundation for a 4D cellular radiotherapy simulation tool.

METHODS

The model, developed in Matlab, contains cell positions randomised in 3D space without overlap. Blood vessels are represented by strings of blood vessel units which branch outwards to achieve the desired tumour relative vascular volume. Hypoxic cells have an increased cell cycle time and become quiescent at oxygen tensions less than 1 mmHg. Necrotic cells are resorbed. A hierarchy of stem cells, transit cells and differentiated cells is considered along with differentiated cell loss. Model parameters include the relative vascular volume (2-10%), blood oxygenation (20-100 mmHg), distance from vessels to the onset of necrosis (80-300 μm) and probability for stem cells to undergo symmetric division (2%). Simulations were performed to observe the effects of hypoxia on tumour growth rate for head and neck cancers. Simulations were run on a supercomputer with eligible parts running in parallel on 12 cores.

RESULTS

Using biologically plausible model parameters for head and neck cancers, the tumour volume doubling time varied from 45 ± 5 days (n = 3) for well oxygenated tumours to 87 ± 5 days (n = 3) for severely hypoxic tumours.

CONCLUSIONS

The main achievements of the current model were randomised cell positions and the connected vasculature structure between the cells. These developments will also be beneficial when irradiating the simulated tumours using Monte Carlo track structure methods.

Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis

We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.

Spatial Metrics of Tumour Vascular Organisation Predict Radiation Efficacy in a Computational Model

Intratumoural heterogeneity is known to contribute to poor therapeutic response. Variations in oxygen tension in particular have been correlated with changes in radiation response in vitro and at the clinical scale with overall survival. Heterogeneity at the microscopic scale in tumour blood vessel architecture has been described, and is one source of the underlying variations in oxygen tension. We seek to determine whether histologic scale measures of the erratic distribution of blood vessels within a tumour can be used to predict differing radiation response. Using a two-dimensional hybrid cellular automaton model of tumour growth, we evaluate the effect of vessel distribution on cell survival outcomes of simulated radiation therapy. Using the standard equations for the oxygen enhancement ratio for cell survival probability under differing oxygen tensions, we calculate average radiation effect over a range of different vessel densities and organisations. We go on to quantify the vessel distribution heterogeneity and measure spatial organization using Ripley’s L function, a measure designed to detect deviations from complete spatial randomness. We find that under differing regimes of vessel density the correlation coefficient between the measure of spatial organization and radiation effect changes sign. This provides not only a useful way to understand the differences seen in radiation effect for tissues based on vessel architecture, but also an alternate explanation for the vessel normalization hypothesis.

In this paper we use a mathematical model, called a hybrid cellular automaton, to study the effect of different vessel distributions on radiation therapy outcomes at the cellular level. We show that the correlation between radiation outcome and spatial organization of vessels changes signs between relatively low and high vessel density. Specifically, that for relatively low vessel density, radiation efficacy is decreased when vessels are more homogeneously distributed, and the opposite is true, that radiation efficacy is improved, when vessel organisation is normalised in high densities. This result suggests an alteration to the vessel normalization hypothesis which states that normalisation of vascular beds should improve radio- and chemo-therapeutic response, but has failed to be validated in clinical studies. In this alteration, we show that Ripley’s L function allows discrimination between vascular architectures in different density regimes in which the standard hypothesis holds and does not hold. Further, we find that this information can be used to augment quantitative histologic analysis of tumours to aid radiation dose personalisation.

From invasion to latency: intracellular noise and cell motility as key controls of the competition between resource-limited cellular populations

In this paper we analyse stochastic models of the competition between two resource-limited cell populations which differ in their response to nutrient availability: the resident population exhibits a switch-like response behaviour while the invading population exhibits a bistable response. We investigate how noise in the intracellular regulatory pathways and cell motility influence the fate of the incumbent and invading populations. We focus initially on a spatially homogeneous system and study in detail the role of intracellular noise. We show that in such well-mixed systems, two distinct regimes exist: In the low (intracellular) noise limit, the invader has the ability to invade the resident population, whereas in the high noise regime competition between the two populations is found to be neutral and, in accordance with neutral evolution theory, invasion is a random event. Careful examination of the system dynamics leads us to conclude that (i) even if the invader is unable to invade, the distribution of survival times, PS(t), has a fat-tail behaviour (PS(t) ∼ t(-1)) which implies that small colonies of mutants can coexist with the resident population for arbitrarily long times, and (ii) the bistable structure of the invading population increases the stability of the latent population, thus increasing their long-term likelihood of survival, by decreasing the intensity of the noise at the population level. We also examine the effects of spatial inhomogeneity. In the low noise limit we find that cell motility is positively correlated with the aggressiveness of the invader as defined by the time the invader takes to invade the resident population: the faster the invasion, the more aggressive the invader.

Using mathematical models to understand metabolism, genes, and disease

Mathematical models are a useful tool for investigating a large number of questions in metabolism, genetics, and gene–environment interactions. A model based on the underlying biology and biochemistry is a platform for in silico biological experimentation that can reveal the causal chain of events that connect variation in one quantity to variation in another. We discuss how we construct such models, how we have used them to investigate homeostatic mechanisms, gene–environment interactions, and genotype–phenotype mapping, and how they can be used in precision and personalized medicine.

Comparing stochastic differential equations and agent-based modelling and simulation for early-stage cancer

There is great potential to be explored regarding the use of agent-based modelling and simulation as an alternative paradigm to investigate early-stage cancer interactions with the immune system. It does not suffer from some limitations of ordinary differential equation models, such as the lack of stochasticity, representation of individual behaviours rather than aggregates and individual memory. In this paper we investigate the potential contribution of agent-based modelling and simulation when contrasted with stochastic versions of ODE models using early-stage cancer examples. We seek answers to the following questions: (1) Does this new stochastic formulation produce similar results to the agent-based version? (2) Can these methods be used interchangeably? (3) Do agent-based models outcomes reveal any benefit when compared to the Gillespie results? To answer these research questions we investigate three well-established mathematical models describing interactions between tumour cells and immune elements. These case studies were re-conceptualised under an agent-based perspective and also converted to the Gillespie algorithm formulation. Our interest in this work, therefore, is to establish a methodological discussion regarding the usability of different simulation approaches, rather than provide further biological insights into the investigated case studies. Our results show that it is possible to obtain equivalent models that implement the same mechanisms; however, the incapacity of the Gillespie algorithm to retain individual memory of past events affects the similarity of some results. Furthermore, the emergent behaviour of ABMS produces extra patters of behaviour in the system, which was not obtained by the Gillespie algorithm.

Targeting the biophysical properties of the myeloma initiating cell niches: a pharmaceutical synergism analysis using multi-scale agent-based modeling

Multiple myeloma, the second most common hematological cancer, is currently incurable due to refractory disease relapse and development of multiple drug resistance. We and others recently established the biophysical model that myeloma initiating (stem) cells (MICs) trigger the stiffening of their niches via SDF-1/CXCR4 paracrine; The stiffened niches then promote the colonogenesis of MICs and protect them from drug treatment. In this work we examined in silico the pharmaceutical potential of targeting MIC niche stiffness to facilitate cytotoxic chemotherapies. We first established a multi-scale agent-based model using the Markov Chain Monte Carlo approach to recapitulate the niche stiffness centric, pro-oncogenetic positive feedback loop between MICs and myeloma-associated bone marrow stromal cells (MBMSCs), and investigated the effects of such intercellular chemo-physical communications on myeloma development. Then we used AMD3100 (to interrupt the interactions between MICs and their stroma) and Bortezomib (a recently developed novel therapeutic agent) as representative drugs to examine if the biophysical properties of myeloma niches are drugable. Results showed that our model recaptured the key experimental observation that the MBMSCs were more sensitive to SDF-1 secreted by MICs, and provided stiffer niches for these initiating cells and promoted their proliferation and drug resistance. Drug synergism analysis suggested that AMD3100 treatment undermined the capability of MICs to modulate the bone marrow microenvironment, and thus re-sensitized myeloma to Bortezomib treatments. This work is also the first attempt to virtually visualize in 3D the dynamics of the bone marrow stiffness during myeloma development. In summary, we established a multi-scale model to facilitate the translation of the niche-stiffness centric myeloma model as well as experimental observations to possible clinical applications. We concluded that targeting the biophysical properties of stem cell niches is of high clinical potential since it may re-sensitize tumor initiating cells to chemotherapies and reduce risks of cancer relapse.

On-lattice agent-based simulation of populations of cells within the open-source Chaste framework

Over the years, agent-based models have been developed that combine cell division and reinforced random walks of cells on a regular lattice, reaction-diffusion equations for nutrients and growth factors; and ordinary differential equations for the subcellular networks regulating the cell cycle. When linked to a vascular layer, this multiple scale model framework has been applied to tumour growth and therapy. Here, we report on the creation of an agent-based multi-scale environment amalgamating the characteristics of these models within a Virtual Physiological Human (VPH) Exemplar Project. This project enables reuse, integration, expansion and sharing of the model and relevant data. The agent-based and reaction-diffusion parts of the multi-scale model have been implemented and are available for download as part of the latest public release of Chaste (Cancer, Heart and Soft Tissue Environment; http://www.cs.ox.ac.uk/chaste/), part of the VPH Toolkit (http://toolkit.vph-noe.eu/). The environment functionalities are verified against the original models, in addition to extra validation of all aspects of the code. In this work, we present the details of the implementation of the agent-based environment, including the system description, the conceptual model, the development of the simulation model and the processes of verification and validation of the simulation results. We explore the potential use of the environment by presenting exemplar applications of the 'what if' scenarios that can easily be studied in the environment. These examples relate to tumour growth, cellular competition for resources and tumour responses to hypoxia (low oxygen levels). We conclude our work by summarizing the future steps for the expansion of the current system.

A mathematical model of HiF-1α-mediated response to hypoxia on the G1/S transition

Hypoxia is known to influence the cell cycle by increasing the G1 phase duration or by inducing a quiescent state (arrest of cell proliferation). This entry into quiescence is a mean for the cell to escape from hypoxia-induced apoptosis. It is suggested that some cancer cells have gain the advantage over normal cells to easily enter into quiescence when environmental conditions, such as oxygen pressure, are unfavorable [43,1]. This ability contributes in the appearance of highly resistant and aggressive tumor phenotypes [2]. The HiF-1α factor is the key actor of the intracellular hypoxia pathway. As tumor cells undergo chronic hypoxic conditions, HiF-1α is present in higher level in cancer than in normal cells. Besides, it was shown that genetic mutations promoting overstabilization of HiF-1α are a feature of various types of cancers [7]. Finally, it is suggested that the intracellular level of HiF-1α can be related to the aggressiveness of the tumors [53,24,4,10]. However, up to now, mathematical models describing the G1/S transition under hypoxia, did not take into account the HiF-1α factor in the hypoxia pathway. Therefore, we propose a mathematical model of the G1/S transition under hypoxia, which explicitly integrates the HiF-1α pathway. The model reproduces the slowing down of G1 phase under moderate hypoxia, and the entry into quiescence of proliferating cells under severe hypoxia. We show how the inhibition of cyclin D by HiF-1α can induce quiescence; this result provides a theoretical explanation to the experimental observations of Wen et al. (2010) [50]. Thus, our model confirms that hypoxia-induced chemoresistance can be linked, for a part, to the negative regulation of cyclin D by HiF-1α.

Evolution and morphology of microenvironment-enhanced malignancy of three-dimensional invasive solid tumors

The emergence of invasive and metastatic behavior in malignant tumors can often lead to fatal outcomes for patients. The collective malignant tumor behavior resulting from the complex tumor-host interactions and the interactions between the tumor cells is currently poorly understood. In this paper, we employ a cellular automaton (CA) model to investigate microenvironment-enhanced malignant behaviors and morphologies of in vitro avascular invasive solid tumors in three dimensions. Our CA model incorporates a variety of microscopic-scale tumor-host interactions, including the degradation of the extracellular matrix by the malignant cells, nutrient-driven cell migration, pressure buildup due to the deformation of the microenvironment by the growing tumor, and its effect on the local tumor-host interface stability. Moreover, the effects of cell-cell adhesion on tumor growth are explicitly taken into account. Specifically, we find that while strong cell-cell adhesion can suppress the invasive behavior of the tumors growing in soft microenvironments, cancer malignancy can be significantly enhanced by harsh microenvironmental conditions, such as exposure to high pressure levels. We infer from the simulation results a qualitative phase diagram that characterizes the expected malignant behavior of invasive solid tumors in terms of two competing malignancy effects: the rigidity of the microenvironment and cell-cell adhesion. This diagram exhibits phase transitions between noninvasive and invasive behaviors. We also discuss the implications of our results for the diagnosis, prognosis, and treatment of malignant tumors.

Inter-cellular signaling network reveals a mechanistic transition in tumor microenvironment

We conducted inter-cellular cytokine correlation and network analysis based upon a stochastic population dynamics model that comprises five cell types and fifteen signaling molecules inter-connected through a large number of cell-cell communication pathways. We observed that the signaling molecules are tightly correlated even at very early stages (e.g. the first month) of human glioma, but such correlation rapidly diminishes when tumor grows to a size that can be clinically detected. Further analysis suggests that paracrine is shown to be the dominant force during tumor initiation and priming, while autocrine supersedes it and supports a robust tumor expansion. In correspondence, the cytokine correlation network evolves through an increasing to decreasing complexity. This study indicates a possible mechanistic transition from the microenvironment-controlled, paracrine-based regulatory mechanism to self-sustained rapid progression to fetal malignancy. It also reveals key nodes that are responsible for such transition and can be potentially harnessed for the design of new anti-cancer therapies.

Systems biology for molecular life sciences and its impact in biomedicine

Modern systems biology is already contributing to a radical transformation of molecular life sciences and biomedicine, and it is expected to have a real impact in the clinical setting in the next years. In this review, the emergence of systems biology is contextualized with a historic overview, and its present state is depicted. The present and expected future contribution of systems biology to the development of molecular medicine is underscored. Concerning the present situation, this review includes a reflection on the "inflation" of biological data and the urgent need for tools and procedures to make hidden information emerge. Descriptions of the impact of networks and models and the available resources and tools for applying them in systems biology approaches to molecular medicine are provided as well. The actual current impact of systems biology in molecular medicine is illustrated, reviewing two cases, namely, those of systems pharmacology and cancer systems biology. Finally, some of the expected contributions of systems biology to the immediate future of molecular medicine are commented.

Multi-scale agent-based brain cancer modeling and prediction of TKI treatment response: incorporating EGFR signaling pathway and angiogenesis

Background

The epidermal growth factor receptor (EGFR) signaling pathway and angiogenesis in brain cancer act as an engine for tumor initiation, expansion and response to therapy. Since the existing literature does not have any models that investigate the impact of both angiogenesis and molecular signaling pathways on treatment, we propose a novel multi-scale, agent-based computational model that includes both angiogenesis and EGFR modules to study the response of brain cancer under tyrosine kinase inhibitors (TKIs) treatment.

Results

The novel angiogenesis module integrated into the agent-based tumor model is based on a set of reaction–diffusion equations that describe the spatio-temporal evolution of the distributions of micro-environmental factors such as glucose, oxygen, TGFα, VEGF and fibronectin. These molecular species regulate tumor growth during angiogenesis. Each tumor cell is equipped with an EGFR signaling pathway linked to a cell-cycle pathway to determine its phenotype. EGFR TKIs are delivered through the blood vessels of tumor microvasculature and the response to treatment is studied.

Conclusions

Our simulations demonstrated that entire tumor growth profile is a collective behaviour of cells regulated by the EGFR signaling pathway and the cell cycle. We also found that angiogenesis has a dual effect under TKI treatment: on one hand, through neo-vasculature TKIs are delivered to decrease tumor invasion; on the other hand, the neo-vasculature can transport glucose and oxygen to tumor cells to maintain their metabolism, which results in an increase of cell survival rate in the late simulation stages.

The HYP-RT hypoxic tumour radiotherapy algorithm and accelerated repopulation dose per fraction study

The HYP-RT model simulates hypoxic tumour growth for head and neck cancer as well as radiotherapy and the effects of accelerated repopulation and reoxygenation. This report outlines algorithm design, parameterisation and the impact of accelerated repopulation on the increase in dose/fraction needed to control the extra cell propagation during accelerated repopulation. Cell kill probabilities are based on Linear Quadratic theory, with oxygenation levels and proliferative capacity influencing cell death. Hypoxia is modelled through oxygen level allocation based on pO₂ histograms. Accelerated repopulation is modelled by increasing the stem cell symmetrical division probability, while the process of reoxygenation utilises randomised pO₂ increments to the cell population after each treatment fraction. Propagation of 10⁸ tumour cells requires 5–30 minutes. Controlling the extra cell growth induced by accelerated repopulation requires a dose/fraction increase of 0.5–1.0 Gy, in agreement with published reports. The average reoxygenation pO₂ increment of 3 mmHg per fraction results in full tumour reoxygenation after shrinkage to approximately 1 mm. HYP-RT is a computationally efficient model simulating tumour growth and radiotherapy, incorporating accelerated repopulation and reoxygenation. It may be used to explore cell kill outcomes during radiotherapy while varying key radiobiological and tumour specific parameters, such as the degree of hypoxia.

Modelling the effects of cell-cycle heterogeneity on the response of a solid tumour to chemotherapy: biological insights from a hybrid multiscale cellular automaton model

The therapeutic control of a solid tumour depends critically on the responses of the individual cells that constitute the entire tumour mass. A particular cell's spatial location within the tumour and intracellular interactions, including the evolution of the cell-cycle within each cell, has an impact on their decision to grow and divide. They are also influenced by external signals from other cells as well as oxygen and nutrient concentrations. Hence, it is important to take these into account when modelling tumour growth and the response to various treatment regimes ('cell-kill therapies'), including chemotherapy. In order to address this multiscale nature of solid tumour growth and its response to treatment, we propose a hybrid, individual-based approach that analyses spatio-temporal dynamics at the level of cells, linking individual cell behaviour with the macroscopic behaviour of cell organisation and the microenvironment. The individual tumour cells are modelled by using a cellular automaton (CA) approach, where each cell has its own internal cell-cycle, modelled using a system of ODEs. The internal cell-cycle dynamics determine the growth strategy in the CA model, making it more predictive and biologically relevant. It also helps to classify the cells according to their cell-cycle states and to analyse the effect of various cell-cycle dependent cytotoxic drugs. Moreover, we have incorporated the evolution of oxygen dynamics within this hybrid model in order to study the effects of the microenvironment in cell-cycle regulation and tumour treatments. An important factor from the treatment point of view is that the low concentration of oxygen can result in a hypoxia-induced quiescence (G0/G1 arrest) of the cancer cells, making them resistant to key cytotoxic drugs. Using this multiscale model, we investigate the impact of oxygen heterogeneity on the spatio-temporal patterning of the cell distribution and their cell-cycle status. We demonstrate that oxygen transport limitations result in significant heterogeneity in HIF-1 α signalling and cell-cycle status, and when these are combined with drug transport limitations, the efficacy of the therapy is significantly impaired.

Multiscale cancer modeling

Simulating cancer behavior across multiple biological scales in space and time, i.e., multiscale cancer modeling, is increasingly being recognized as a powerful tool to refine hypotheses, focus experiments, and enable more accurate predictions. A growing number of examples illustrate the value of this approach in providing quantitative insights in the initiation, progression, and treatment of cancer. In this review, we introduce the most recent and important multiscale cancer modeling works that have successfully established a mechanistic link between different biological scales. Biophysical, biochemical, and biomechanical factors are considered in these models. We also discuss innovative, cutting-edge modeling methods that are moving predictive multiscale cancer modeling toward clinical application. Furthermore, because the development of multiscale cancer models requires a new level of collaboration among scientists from a variety of fields such as biology, medicine, physics, mathematics, engineering, and computer science, an innovative Web-based infrastructure is needed to support this growing community.

Monte Carlo radiotherapy simulations of accelerated repopulation and reoxygenation for hypoxic head and neck cancer

OBJECTIVE

A temporal Monte Carlo tumour growth and radiotherapy effect model (HYP-RT) simulating hypoxia in head and neck cancer has been developed and used to analyse parameters influencing cell kill during conventionally fractionated radiotherapy. The model was designed to simulate individual cell division up to 10(8) cells, while incorporating radiobiological effects, including accelerated repopulation and reoxygenation during treatment.

METHOD

Reoxygenation of hypoxic tumours has been modelled using randomised increments of oxygen to tumour cells after each treatment fraction. The process of accelerated repopulation has been modelled by increasing the symmetrical stem cell division probability. Both phenomena were onset immediately or after a number of weeks of simulated treatment.

RESULTS

The extra dose required to control (total cell kill) hypoxic vs oxic tumours was 15-25% (8-20 Gy for 5 × 2 Gy per week) depending on the timing of accelerated repopulation onset. Reoxygenation of hypoxic tumours resulted in resensitisation and reduction in total dose required by approximately 10%, depending on the time of onset. When modelled simultaneously, accelerated repopulation and reoxygenation affected cell kill in hypoxic tumours in a similar manner to when the phenomena were modelled individually; however, the degree was altered, with non-additive results. Simulation results were in good agreement with standard linear quadratic theory; however, differed for more complex comparisons where hypoxia, reoxygenation as well as accelerated repopulation effects were considered.

CONCLUSION

Simulations have quantitatively confirmed the need for patient individualisation in radiotherapy for hypoxic head and neck tumours, and have shown the benefits of modelling complex and dynamic processes using Monte Carlo methods.

Adaptation to survival in germinal center is the initial step in onset of indolent stage of multiple myeloma

Aberrant mutations of centrocytes in germinal centers (GC) can generate two completely different diseases: B-cell lymphomas and monoclonal gammopathy of undetermined significance (MGUS). In this article we use computational models to examine the evolutionary dynamics by which initial adaptation to survival in the GC allows naive MGUS cells to proliferate in the bone marrow and initiate the evolutionary process that will lead to aggressive multiple myeloma (MM). Our simulations show that MGUS cells may generate bone marrow tumors ranging from indolent to aggressive, depending on the original adaptation in the GC. All these tumors, however, are limited to approximately 15% of the marrow cellularity due to hypoxia-induced quiescence (this correlates with the cellularity that separates MGUS and MM, ∼10%). Resistance to hypoxia-induced quiescence and cell death was one of the two major bone marrow adaptations that allowed continued tumor growth and establishment of paracrine cytokine loops, known to increase MM cell replication and de novo multidrug resistance. The second major adaptation was an increase in IL-6-independent growth rate, which correlates with the mutations observed in advanced stage patients. Even though there was an increase in the microvessel density in all simulations, the "angiogenic switch" was not due to a MM angiogenic phenotype, but rather the response of MM cells to the regional hypoxia caused by the increased tumor burden. These results indicate that treatments targeting the adaptation to survival and proliferation in hypoxia, in conjunction with currently available therapies, may have synergistic effects, by delaying tumor growth and reducing cytokine paracrine loops mediated by angiogenic factors.

An integrative biological approach to the analysis of tissue culture data: application to the antitumour agent RHPS4

We describe a mathematical model of cell growth and death and explain how it can be used to integrate data from classic tissue culture experiments on antitumour agents and thus aid the identification of their mechanism of action. Experimental data relating to time- and dose-dependent changes in growth rate, cell cycle distribution, plus apoptotic and senescent fractions, are reinterpreted in terms of modulations to kinetic parameters that describe the rates at which cells transit between phenotypic compartments. The mathematical model is analytical, in the sense that the kinetic parameters are calculated from the experimental data directly, without any fitting process. Since the kinetic parameters are much more directly related to potential molecular targets than are the experimentally measured quantities, this approach can provide a more informative picture of the mechanism of action of the antitumour agent under investigation. We demonstrate the potential value of our model by applying it to data from RHPS4 (3,11-difluoro-6,8,13-trimethyl-8H-quino [4,3,2-kl] acridinium methosulfate). This agent is a DNA-interactive pentacyclic acridine for which at least three potential mechanisms of antitumour activity have been identified. Firstly RHPS4 is a telomerase inhibitor, secondly it is a telomerase-independent destabiliser of telomeres, and thirdly it is a telomere-independent binder to genomic DNA. Each mechanism can induce a separate, but overlapping, pattern of cellular responses, making the interpretation of tissue culture data very complex. Here we study the time- and dose-dependent effects of RHPS4 on the HCT116 cell line, and develop a five-compartment mathematical model to interpret the data. Application of the model to the data suggests that RHPS4 increases the rate at which cells became senescent state but, rather surprisingly, actually inhibits the rate of cell death. As a control, we also apply the model to data describing the time- and dose-dependent effects of doxorubicin since its mechanism of action is better characterised.

Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour.

Quiescence: a mechanism for escaping the effects of drug on cell populations

We point out that a simple and generic strategy in order to lower the risk for extinction consists in developing a dormant stage in which the organism is unable to multiply but may die. The dormant organism is protected against the poisonous environment. The result is to increase the survival probability of the entire population by introducing a type of zero reproductive fitness. This is possible, because the reservoir of dormant individuals act as a buffer that can cushion fatal fluctuations in the number of births and deaths, which without the dormant population would have driven the entire population to extinction.

In silico models of cancer

Cancer is a complex disease that involves multiple types of biological interactions across diverse physical, temporal, and biological scales. This complexity presents substantial challenges for the characterization of cancer biology, and motivates the study of cancer in the context of molecular, cellular, and physiological systems. Computational models of cancer are being developed to aid both biological discovery and clinical medicine. The development of these in silico models is facilitated by rapidly advancing experimental and analytical tools that generate information-rich, high-throughput biological data. Statistical models of cancer at the genomic, transcriptomic, and pathway levels have proven effective in developing diagnostic and prognostic molecular signatures, as well as in identifying perturbed pathways. Statistically inferred network models can prove useful in settings where data overfitting can be avoided, and provide an important means for biological discovery. Mechanistically based signaling and metabolic models that apply a priori knowledge of biochemical processes derived from experiments can also be reconstructed where data are available, and can provide insight and predictive ability regarding the behavior of these systems. At longer length scales, continuum and agent-based models of the tumor microenvironment and other tissue-level interactions enable modeling of cancer cell populations and tumor progression. Even though cancer has been among the most-studied human diseases using systems approaches, significant challenges remain before the enormous potential of in silico cancer biology can be fully realized.

Cancer models, genomic instability and somatic cellular Darwinian evolution

The biology of cancer is critically reviewed and evidence adduced that its development can be modelled as a somatic cellular Darwinian evolutionary process. The evidence for involvement of genomic instability (GI) is also reviewed. A variety of quasi-mechanistic models of carcinogenesis are reviewed, all based on this somatic Darwinian evolutionary hypothesis; in particular, the multi-stage model of Armitage and Doll (Br. J. Cancer 1954:8;1-12), the two-mutation model of Moolgavkar, Venzon, and Knudson (MVK) (Math. Biosci. 1979:47;55-77), the generalized MVK model of Little (Biometrics 1995:51;1278-1291) and various generalizations of these incorporating effects of GI (Little and Wright Math. Biosci. 2003:183;111-134; Little et al. J. Theoret. Biol. 2008:254;229-238).

In silico models of cancer

Cancer is a complex disease that involves multiple types of biological interactions across diverse physical, temporal, and biological scales. This complexity presents substantial challenges for the characterization of cancer biology, and motivates the study of cancer in the context of molecular, cellular, and physiological systems. Computational models of cancer are being developed to aid both biological discovery and clinical medicine. The development of these in silico models is facilitated by rapidly advancing experimental and analytical tools that generate information-rich, high-throughput biological data. Statistical models of cancer at the genomic, transcriptomic, and pathway levels have proven effective in developing diagnostic and prognostic molecular signatures, as well as in identifying perturbed pathways. Statistically inferred network models can prove useful in settings where data overfitting can be avoided, and provide an important means for biological discovery. Mechanistically based signaling and metabolic models that apply a priori knowledge of biochemical processes derived from experiments can also be reconstructed where data are available, and can provide insight and predictive ability regarding the behavior of these systems. At longer length scales, continuum and agent-based models of the tumor microenvironment and other tissue-level interactions enable modeling of cancer cell populations and tumor progression. Even though cancer has been among the most-studied human diseases using systems approaches, significant challenges remain before the enormous potential of in silico cancer biology can be fully realized.

Introduction of hypermatrix and operator notation into a discrete mathematics simulation model of malignant tumour response to therapeutic schemes in vivo. Some operator properties

The tremendous rate of accumulation of experimental and clinical knowledge pertaining to cancer dictates the development of a theoretical framework for the meaningful integration of such knowledge at all levels of biocomplexity. In this context our research group has developed and partly validated a number of spatiotemporal simulation models of in vivo tumour growth and in particular tumour response to several therapeutic schemes. Most of the modeling modules have been based on discrete mathematics and therefore have been formulated in terms of rather complex algorithms (e.g. in pseudocode and actual computer code). However, such lengthy algorithmic descriptions, although sufficient from the mathematical point of view, may render it difficult for an interested reader to readily identify the sequence of the very basic simulation operations that lie at the heart of the entire model. In order to both alleviate this problem and at the same time provide a bridge to symbolic mathematics, we propose the introduction of the notion of hypermatrix in conjunction with that of a discrete operator into the already developed models. Using a radiotherapy response simulation example we demonstrate how the entire model can be considered as the sequential application of a number of discrete operators to a hypermatrix corresponding to the dynamics of the anatomic area of interest. Subsequently, we investigate the operators’ commutativity and outline the “summarize and jump” strategy aiming at efficiently and realistically address multilevel biological problems such as cancer. In order to clarify the actual effect of the composite discrete operator we present further simulation results which are in agreement with the outcome of the clinical study RTOG 83–02, thus strengthening the reliability of the model developed.

The influence of P-glycoprotein expression and its inhibitors on the distribution of doxorubicin in breast tumors

Background

Anti-cancer drugs access solid tumors via blood vessels, and must penetrate tumor tissue to reach all cancer cells. Previous studies have demonstrated steep gradients of decreasing doxorubicin fluorescence with increasing distance from blood vessels, such that many tumor cells are not exposed to drug. Studies using multilayered cell cultures show that increased P-glycoprotein (PgP) is associated with better penetration of doxorubicin, while PgP inhibitors decrease drug penetration in tumor tissue. Here we evaluate the effect of PgP expression on doxorubicin distribution in vivo.

Methods

Mice bearing tumor sublines with either high or low expression of PgP were treated with doxorubicin, with or without pre-treatment with the PgP inhibitors verapamil or PSC 833. The distribution of doxorubicin in relation to tumor blood vessels was quantified using immunofluorescence.

Results

Our results indicate greater uptake of doxorubicin by cells near blood vessels in wild type as compared to PgP-overexpressing tumors, and pre-treatment with verapamil or PSC 833 increased uptake in PgP-overexpressing tumors. However, there were steeper gradients of decreasing doxorubicin fluorescence in wild-type tumors compared to PgP overexpressing tumors, and treatment of PgP overexpressing tumors with PgP inhibitors led to steeper gradients and greater heterogeneity in the distribution of doxorubicin.

Conclusion

PgP inhibitors increase uptake of doxorubicin in cells close to blood vessels, have little effect on drug uptake into cells at intermediate distances, and might have a paradoxical effect to decrease doxorubicin uptake into distal cells. This effect probably contributes to the limited success of PgP inhibitors in clinical trials.

Comparing the growth kinetics of cell populations in two and three dimensions

We study the kinetics of growing cell populations by means of a kinetic Monte Carlo method. By applying the same growth mechanism to a two-dimensional (2D) and a three-dimensional (3D) model, and making direct comparison with experimental studies, we show that both models exhibit similar behavior. Based on this we propose a method for establishment of a mapping between the 2D and 3D results. Additionally, we present an analytic approach to obtain the time evolution, and show in case of the 3D model how synchronization effects can influence the growth kinetics. Finally, we compare the results of our models to experimental data of the growth kinetics of 2D monolayers and 3D NIH3T3 xenografts in mice.