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

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.

Bystander effects and their implications for clinical radiation therapy: Insights from multiscale in silico experiments

Radiotherapy is a commonly used treatment for cancer and is usually given in varying doses. At low radiation doses relatively few cells die as a direct response to radiation but secondary radiation effects, such as DNA mutation or bystander phenomena, may affect many cells. Consequently it is at low radiation levels where an understanding of bystander effects is essential in designing novel therapies with superior clinical outcomes. In this paper, we use a hybrid multiscale mathematical model to study the direct effects of radiation as well as radiation-induced bystander effects on both tumour cells and normal cells. We show that bystander responses play a major role in mediating radiation damage to cells at low-doses of radiotherapy, doing more damage than that due to direct radiation. The survival curves derived from our computational simulations showed an area of hyper-radiosensitivity at low-doses that are not obtained using a traditional radiobiological model.

Optimal control strategies of eradicating invisible glioblastoma cells after conventional surgery

Glioblastoma, the most aggressive type of brain cancer, has median survival time of 1 year after diagnosis. It is characterized by alternating modes of rapid proliferation and aggressive invasion in response to metabolic stress in the microenvironment. A particular microRNA, miR-451, and its downstream signalling molecules, AMPK complex, are known to be key determinants in switching cell fate. These components form a core control system determining a balance between cell growth and migration which is regulated by fluctuating glucose levels in the microenvironment. An important factor from the treatment point of view is that low levels of glucose affect metabolism and activate cell migration through the miR-451-AMPK control system, creating 'invisible' migratory cells and making them inaccessible by conventional surgery. In this work, we apply optimal control theory to deal with the problem of maintaining upregulated miR-451 levels that prevent cell infiltration to surrounding brain tissue and thus induce localization of these cancer cells at the surgical site. The model also considers the effect of a drug that blocks inhibitive pathways of miR-451 from AMPK complex. Glucose infusion control and drug infusion control are chosen to represent dose rates of glucose and drug intravenous administrations, respectively. The characteristics of optimal control lead us to investigate the structure of optimal intravenous infusion regimen under various circumstances and predict best clinical outcomes with minimum expense possible.

Microenvironmental Niches and Sanctuaries: A Route to Acquired Resistance

A tumor vasculature that is functionally abnormal results in irregular gradients of metabolites and drugs within the tumor tissue. Recently, significant efforts have been committed to experimentally examine how cellular response to anti-cancer treatments varies based on the environment in which the cells are grown. In vitro studies point to specific conditions in which tumor cells can remain dormant and survive the treatment. In vivo results suggest that cells can escape the effects of drug therapy in tissue regions that are poorly penetrated by the drugs. Better understanding how the tumor microenvironments influence the emergence of drug resistance in both primary and metastatic tumors may improve drug development and the design of more effective therapeutic protocols. This chapter presents a hybrid agent-based model of the growth of tumor micrometastases and explores how microenvironmental factors can contribute to the development of acquired resistance in response to a DNA damaging drug. The specific microenvironments of interest in this work are tumor hypoxic niches and tumor normoxic sanctuaries with poor drug penetration. We aim to quantify how spatial constraints of limited drug transport and quiescent cell survival contribute to the development of drug resistant tumors.

Spatial Modeling of Drug Delivery Routes for Treatment of Disseminated Ovarian Cancer

In ovarian cancer, metastasis is typically confined to the peritoneum. Surgical removal of the primary tumor and macroscopic secondary tumors is a common practice, but more effective strategies are needed to target microscopic spheroids persisting in the peritoneal fluid after debulking surgery. To treat this residual disease, therapeutic agents can be administered by either intravenous or intraperitoneal infusion. Here, we describe the use of a cellular Potts model to compare tumor penetration of two classes of drugs (cisplatin and pertuzumab) when delivered by these two alternative routes. The model considers the primary route when the drug is administered either intravenously or intraperitoneally, as well as the subsequent exchange into the other delivery volume as a secondary route. By accounting for these dynamics, the model revealed that intraperitoneal infusion is the markedly superior route for delivery of both small-molecule and antibody therapies into microscopic, avascular tumors typical of patients with ascites. Small tumors attached to peritoneal organs, with vascularity ranging from 2% to 10%, also show enhanced drug delivery via the intraperitoneal route, even though tumor vessels can act as sinks during the dissemination of small molecules. Furthermore, we assessed the ability of the antibody to enter the tumor by in silico and in vivo methods and suggest that optimization of antibody delivery is an important criterion underlying the efficacy of these and other biologics. The use of both delivery routes may provide the best total coverage of tumors, depending on their size and vascularity.

Model-Based Tumor Growth Dynamics and Therapy Response in a Mouse Model of De Novo Carcinogenesis

Tumorigenesis is a complex, multistep process that depends on numerous alterations within the cell and contribution from the surrounding stroma. The ability to model macroscopic tumor evolution with high fidelity may contribute to better predictive tools for designing tumor therapy in the clinic. However, attempts to model tumor growth have mainly been developed and validated using data from xenograft mouse models, which fail to capture important aspects of tumorigenesis including tumor-initiating events and interactions with the immune system. In the present study, we investigate tumor growth and therapy dynamics in a mouse model of de novo carcinogenesis that closely recapitulates tumor initiation, progression and maintenance in vivo. We show that the rate of tumor growth and the effects of therapy are highly variable and mouse specific using a Gompertz model to describe tumor growth and a two-compartment pharmacokinetic/ pharmacodynamic model to describe the effects of therapy in mice treated with 5-FU. We show that inter-mouse growth variability is considerably larger than intra-mouse variability and that there is a correlation between tumor growth and drug kill rates. Our results show that in vivo tumor growth and regression in a double transgenic mouse model are highly variable both within and between subjects and that mathematical models can be used to capture the overall characteristics of this variability. In order for these models to become useful tools in the design of optimal therapy strategies and ultimately in clinical practice, a subject-specific modelling strategy is necessary, rather than approaches that are based on the average behavior of a given subject population which could provide erroneous results.

A mathematical model of pre-diagnostic glioma growth

Due to their location, the malignant gliomas of the brain in humans are very difficult to treat in advanced stages. Blood-based biomarkers for glioma are needed for more accurate evaluation of treatment response as well as early diagnosis. However, biomarker research in primary brain tumors is challenging given their relative rarity and genetic diversity. It is further complicated by variations in the permeability of the blood brain barrier that affects the amount of marker released into the bloodstream. Inspired by recent temporal data indicating a possible decrease in serum glucose levels in patients with gliomas yet to be diagnosed, we present an ordinary differential equation model to capture early stage glioma growth. The model contains glioma-glucose-immune interactions and poses a potential mechanism by which this glucose drop can be explained. We present numerical simulations, parameter sensitivity analysis, linear stability analysis and a numerical experiment whereby we show how a dormant glioma can become malignant.

Spatio-Temporal Dynamics of Hypoxia during Radiotherapy

Tumour hypoxia plays a pivotal role in cancer therapy for most therapeutic approaches from radiotherapy to immunotherapy. The detailed and accurate knowledge of the oxygen distribution in a tumour is necessary in order to determine the right treatment strategy. Still, due to the limited spatial and temporal resolution of imaging methods as well as lacking fundamental understanding of internal oxygenation dynamics in tumours, the precise oxygen distribution map is rarely available for treatment planing. We employ an agent-based in silico tumour spheroid model in order to study the complex, localized and fast oxygen dynamics in tumour micro-regions which are induced by radiotherapy. A lattice-free, 3D, agent-based approach for cell representation is coupled with a high-resolution diffusion solver that includes a tissue density-dependent diffusion coefficient. This allows us to assess the space- and time-resolved reoxygenation response of a small subvolume of tumour tissue in response to radiotherapy. In response to irradiation the tumour nodule exhibits characteristic reoxygenation and re-depletion dynamics which we resolve with high spatio-temporal resolution. The reoxygenation follows specific timings, which should be respected in treatment in order to maximise the use of the oxygen enhancement effects. Oxygen dynamics within the tumour create windows of opportunity for the use of adjuvant chemotherapeutica and hypoxia-activated drugs. Overall, we show that by using modelling it is possible to follow the oxygenation dynamics beyond common resolution limits and predict beneficial strategies for therapy and in vitro verification. Models of cell cycle and oxygen dynamics in tumours should in the future be combined with imaging techniques, to allow for a systematic experimental study of possible improved schedules and to ultimately extend the reach of oxygenation monitoring available in clinical treatment.

Pathology to enhance precision medicine in oncology: lessons from landscape ecology

A major goal of modern medicine is increasing patient specificity so that the right treatment is administered to the right patient at the right time with the right dose. While current cancer studies have largely focused on identification of genetic or epigenetic properties of tumor cells, emerging evidence has clearly demonstrated substantial genetic heterogeneity between tumors in the same patient and within subclones of a single tumor. Thus, molecular analysis from populations of cells (either a whole tumor or small biopsy of that tumor) is, at best, an incomplete representation of the underlying biology. These observations indicate a significant need to define intratumoral evolutionary dynamics that yield the observed spatial variations in cellular properties. It is generally accepted that genetic heterogeneity among cancer cells is a manifestation of intratumoral evolution, and this is typically viewed as a consequence of random mutations generated by genomic instability within the cancer cells. We suggest that this represents an incomplete view of Darwinian dynamics, which typically are governed by phenotypic variations in response to spatial and temporal heterogeneity in environmental selection forces. We propose that pathologic feature analysis can provide precise information regarding regional variations in environmental selection forces and phenotypic adaptations. These observations can be integrated using quantitative, spatially explicit methods developed in landscape ecology to interrogate heterogenous biological processes in tumors within individual patients. The ability to investigate tumor heterogeneity has been shown to inform physicians regarding critical aspects of cancer progression including invasion, metastasis, drug resistance, and disease relapse.

Strategies of eradicating glioma cells: a multi-scale mathematical model with MiR-451-AMPK-mTOR control

The cellular dispersion and therapeutic control of glioblastoma, the most aggressive type of primary brain cancer, depends critically on the migration patterns after surgery and intracellular responses of the individual cancer cells in response to external biochemical and biomechanical cues in the microenvironment. Recent studies have shown that a particular microRNA, miR-451, regulates downstream molecules including AMPK and mTOR to determine the balance between rapid proliferation and invasion in response to metabolic stress in the harsh tumor microenvironment. Surgical removal of main tumor is inevitably followed by recurrence of the tumor due to inaccessibility of dispersed tumor cells in normal brain tissue. In order to address this multi-scale nature of glioblastoma proliferation and invasion and its response to conventional treatment, we propose a hybrid model of glioblastoma that analyses spatio-temporal dynamics at the cellular level, linking individual tumor cells with the macroscopic behaviour of cell organization and the microenvironment, and with the intracellular dynamics of miR-451-AMPK-mTOR signaling within a tumour cell. The model identifies a key mechanism underlying the molecular switches between proliferative phase and migratory phase in response to metabolic stress and biophysical interaction between cells in response to fluctuating glucose levels in the presence of blood vessels (BVs). The model predicts that cell migration, therefore efficacy of the treatment, not only depends on oxygen and glucose availability but also on the relative balance between random motility and strength of chemoattractants. Effective control of growing cells near BV sites in addition to relocalization of invisible migratory cells back to the resection site was suggested as a way of eradicating these migratory cells.

Integrated PK-PD and agent-based modeling in oncology

Mathematical modeling has become a valuable tool that strives to complement conventional biomedical research modalities in order to predict experimental outcome, generate new medical hypotheses, and optimize clinical therapies. Two specific approaches, pharmacokinetic-pharmacodynamic (PK-PD) modeling, and agent-based modeling (ABM), have been widely applied in cancer research. While they have made important contributions on their own (e.g., PK-PD in examining chemotherapy drug efficacy and resistance, and ABM in describing and predicting tumor growth and metastasis), only a few groups have started to combine both approaches together in an effort to gain more insights into the details of drug dynamics and the resulting impact on tumor growth. In this review, we focus our discussion on some of the most recent modeling studies building on a combined PK-PD and ABM approach that have generated experimentally testable hypotheses. Some future directions are also discussed.

A matter of timing: identifying significant multi-dose radiotherapy improvements by numerical simulation and genetic algorithm search

Multi-dose radiotherapy protocols (fraction dose and timing) currently used in the clinic are the product of human selection based on habit, received wisdom, physician experience and intra-day patient timetabling. However, due to combinatorial considerations, the potential treatment protocol space for a given total dose or treatment length is enormous, even for relatively coarse search; well beyond the capacity of traditional in-vitro methods. In constrast, high fidelity numerical simulation of tumor development is well suited to the challenge. Building on our previous single-dose numerical simulation model of EMT6/Ro spheroids, a multi-dose irradiation response module is added and calibrated to the effective dose arising from 18 independent multi-dose treatment programs available in the experimental literature. With the developed model a constrained, non-linear, search for better performing cadidate protocols is conducted within the vicinity of two benchmarks by genetic algorithm (GA) techniques. After evaluating less than 0.01% of the potential benchmark protocol space, candidate protocols were identified by the GA which conferred an average of 9.4% (max benefit 16.5%) and 7.1% (13.3%) improvement (reduction) on tumour cell count compared to the two benchmarks, respectively. Noticing that a convergent phenomenon of the top performing protocols was their temporal synchronicity, a further series of numerical experiments was conducted with periodic time-gap protocols (10 h to 23 h), leading to the discovery that the performance of the GA search candidates could be replicated by 17-18 h periodic candidates. Further dynamic irradiation-response cell-phase analysis revealed that such periodicity cohered with latent EMT6/Ro cell-phase temporal patterning. Taken together, this study provides powerful evidence towards the hypothesis that even simple inter-fraction timing variations for a given fractional dose program may present a facile, and highly cost-effecitive means of significantly improving clinical efficacy.

Simulating cancer growth with multiscale agent-based modeling

There have been many techniques developed in recent years to in silico model a variety of cancer behaviors. Agent-based modeling is a specific discrete-based hybrid modeling approach that allows simulating the role of diversity in cell populations as well as within each individual cell; it has therefore become a powerful modeling method widely used by computational cancer researchers. Many aspects of tumor morphology including phenotype-changing mutations, the adaptation to microenvironment, the process of angiogenesis, the influence of extracellular matrix, reactions to chemotherapy or surgical intervention, the effects of oxygen and nutrient availability, and metastasis and invasion of healthy tissues have been incorporated and investigated in agent-based models. In this review, we introduce some of the most recent agent-based models that have provided insight into the understanding of cancer growth and invasion, spanning multiple biological scales in time and space, and we further describe several experimentally testable hypotheses generated by those models. We also discuss some of the current challenges of multiscale agent-based cancer models.

The formation of tight tumor clusters affects the efficacy of cell cycle inhibitors: a hybrid model study

Cyclin-dependent kinases (CDKs) are vital in regulating cell cycle progression, and, thus, in highly proliferating tumor cells CDK inhibitors are gaining interest as potential anticancer agents. Clonogenic assay experiments are frequently used to determine drug efficacy against the survival and proliferation of cancer cells. While the anticancer mechanisms of drugs are usually described at the intracellular single-cell level, the experimental measurements are sampled from the entire cancer cell population. This approach may lead to discrepancies between the experimental observations and theoretical explanations of anticipated drug mechanisms. To determine how individual cell responses to drugs that inhibit CDKs affect the growth of cancer cell populations, we developed a spatially explicit hybrid agent-based model. In this model, each cell is equipped with internal cell cycle regulation mechanisms, but it is also able to interact physically with its neighbors. We model cell cycle progression, focusing on the G1 and G2/M cell cycle checkpoints, as well as on related essential components, such as CDK1, CDK2, cell size, and DNA damage. We present detailed studies of how the emergent properties (e.g., cluster formation) of an entire cell population depend on altered physical and physiological parameters. We analyze the effects of CDK1 and CKD2 inhibitors on population growth, time-dependent changes in cell cycle distributions, and the dynamic evolution of spatial cell patterns. We show that cell cycle inhibitors that cause cell arrest at different cell cycle phases are not necessarily synergistically super-additive. Finally, we demonstrate that the physical aspects of cell population growth, such as the formation of tight cell clusters versus dispersed colonies, alter the efficacy of cell cycle inhibitors, both in 2D and 3D simulations. This finding may have implications for interpreting the treatment efficacy results of in vitro experiments, in which treatment is applied before the cells can grow to produce clusters, especially because in vivo tumors, in contrast, form large masses before they are detected and treated.

The effect of the G1-S transition checkpoint on an age structured cell cycle model

Knowledge of how a population of cancerous cells progress through the cell cycle is vital if the population is to be treated effectively, as treatment outcome is dependent on the phase distributions of the population. Estimates on the phase distribution may be obtained experimentally however the errors present in these estimates may effect treatment efficacy and planning. If mathematical models are to be used to make accurate, quantitative predictions concerning treatments, whose efficacy is phase dependent, knowledge of the phase distribution is crucial. In this paper it is shown that two different transition rates at the G1-S checkpoint provide a good fit to a growth curve obtained experimentally. However, the different transition functions predict a different phase distribution for the population, but both lying within the bounds of experimental error. Since treatment outcome is effected by the phase distribution of the population this difference may be critical in treatment planning. Using an age-structured population balance approach the cell cycle is modelled with particular emphasis on the G1-S checkpoint. By considering the probability of cells transitioning at the G1-S checkpoint, different transition functions are obtained. A suitable finite difference scheme for the numerical simulation of the model is derived and shown to be stable. The model is then fitted using the different probability transition functions to experimental data and the effects of the different probability transition functions on the model's results are discussed.

Upregulation of the Kank1 gene-induced brain glioma apoptosis and blockade of the cell cycle in G0/G1 phase

The Kank1 gene is one of the important members of the Kank gene family. As an important adaptor protein, Kank1 plays a significant role in the genesis and development of many malignant tumors. It was recently discovered that the Kank1 gene is a new cancer suppressor, and its expression is significantly downregulated or it is not expressed in kidney cancer, bladder cancer, prostate cancer, lung cancer and breast cancer. However, no report on the role of Kank1 in the genesis of brain glioma is available to date. In this study, we found significantly lower expression of the Kank1 gene in human brain glioma cells compared to the other cells evaluated. We used RNA interference techniques to silence Kank1 gene expression and found acceleration of tumor cell proliferation. However, when the Kank1 gene was upregulated, cell apoptosis occurred and the cell cycle was blocked in the G0/G1 phase. Also, we found that upregulating the Kank1 gene may result in the change of mitochondrial membrane potential, and the regulation of Bax and Bcl-2 may promote the mitochondria to release cytochrome C so as to activate Caspase-9 and -3. Thus, the human brain glioma apoptosis induced by upregulation of the Kank1 gene is closely relevant to the mitochondrial pathway.

A High-Performance Cellular Automaton Model of Tumor Growth with Dynamically Growing Domains

Tumor growth from a single transformed cancer cell up to a clinically apparent mass spans many spatial and temporal orders of magnitude. Implementation of cellular automata simulations of such tumor growth can be straightforward but computing performance often counterbalances simplicity. Computationally convenient simulation times can be achieved by choosing appropriate data structures, memory and cell handling as well as domain setup. We propose a cellular automaton model of tumor growth with a domain that expands dynamically as the tumor population increases. We discuss memory access, data structures and implementation techniques that yield high-performance multi-scale Monte Carlo simulations of tumor growth. We discuss tumor properties that favor the proposed high-performance design and present simulation results of the tumor growth model. We estimate to which parameters the model is the most sensitive, and show that tumor volume depends on a number of parameters in a non-monotonic manner.

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.

Mathematical models for translational and clinical oncology

In the context of translational and clinical oncology, mathematical models can provide novel insights into tumor-related processes and can support clinical oncologists in the design of the treatment regime, dosage, schedule, toxicity and drug-sensitivity. In this review we present an overview of mathematical models in this field beginning with carcinogenesis and proceeding to the different cancer treatments. By doing so we intended to highlight recent developments and emphasize the power of such theoretical work.We first highlight mathematical models for translational oncology comprising epidemiologic and statistical models, mechanistic models for carcinogenesis and tumor growth, as well as evolutionary dynamics models which can help to describe and overcome a major problem in the clinic: therapy resistance. Next we review models for clinical oncology with a special emphasis on therapy including chemotherapy, targeted therapy, radiotherapy, immunotherapy and interaction of cancer cells with the immune system.As evident from the published studies, mathematical modeling and computational simulation provided valuable insights into the molecular mechanisms of cancer, and can help to improve diagnosis and prognosis of the disease, and pinpoint novel therapeutic targets.

How to build a multiscale model in biology

Biological processes span several scales in space, from the single molecules to organisms and ecosystems. Multiscale modelling approaches in biology are useful to take into account the complex interactions between different organisation levels in those systems. We review several single- and multiscale models, from the most simple to the complex ones, and discuss their properties from a multiscale point of view. Approaches based on master equations for stochastic processes, individual-based models, hybrid continuous-discrete models and structured PDE models are presented.

Towards predicting the response of a solid tumour to chemotherapy and radiotherapy treatments: clinical insights from a computational model

In this paper we use a hybrid multiscale mathematical model that incorporates both individual cell behaviour through the cell-cycle and the effects of the changing microenvironment through oxygen dynamics to study the multiple effects of radiation therapy. The oxygenation status of the cells is considered as one of the important prognostic markers for determining radiation therapy, as hypoxic cells are less radiosensitive. Another factor that critically affects radiation sensitivity is cell-cycle regulation. The effects of radiation therapy are included in the model using a modified linear quadratic model for the radiation damage, incorporating the effects of hypoxia and cell-cycle in determining the cell-cycle phase-specific radiosensitivity. Furthermore, after irradiation, an individual cell's cell-cycle dynamics are intrinsically modified through the activation of pathways responsible for repair mechanisms, often resulting in a delay/arrest in the cell-cycle. The model is then used to study various combinations of multiple doses of cell-cycle dependent chemotherapies and radiation therapy, as radiation may work better by the partial synchronisation of cells in the most radiosensitive phase of the cell-cycle. Moreover, using this multi-scale model, we investigate the optimum sequencing and scheduling of these multi-modality treatments, and the impact of internal and external heterogeneity on the spatio-temporal patterning of the distribution of tumour cells and their response to different treatment schedules.

Anti-cancer treatments such as radiotherapy and chemotherapy have evolved through clinical trial-and-error over decades, and although they cure some cases and are partially effective in many, the majority of such cancers ultimately recur. Doctors turn to new, expensive drugs as they emerge, but perhaps fail to study and learn how to use the therapies they already have most effectively. This is partly because clinical trials are expensive to conduct, both in terms of time and money. The cancer cell is complicated, but many mechanisms that control its response to treatment are now understood. We show here how a mathematical model accurately reproduces the results of previous biological experiments of cancer treatment, opening up the possibility of using it to predict which combinations of drugs and radiotherapy would be best for patients.

Acid-mediated tumour cell invasion: a discrete modelling approach using the extended Potts model

Acidic extracellular pH has been shown to play a crucial part in the invasive and metastatic cascade of some tumours. In this study, we examine the effect of extracellular acidity on tumour invasion focusing, in particular, on cellular adhesion, proteolytic enzyme activity and cellular proliferation. Our numerical simulations using a cellular Potts model show that, under acidic extracellular pH, changes in cell-matrix adhesion strength has a comparable effect on tumour invasiveness as the increase in proteolytic enzyme activity. We also show that tumour cells cultured under physiological pH tend to be large and the tumours develop a "diffuse" morphology compared to those cultured at acidic pH, which display protruding "fingers" at the advancing front. A key model prediction is the observation that the main effect on invasion from culturing cells at low extracellular pH stems from changes in the intercellular and cell-matrix adhesion strengths and proteolytic enzyme secretion rate. However, we show that the effects of proteolysis needs to be significant as low to moderate changes only has nominal effects on cell invasiveness. We find that the low pH e effects on cell size and proliferation rate have much lower influence on cell invasiveness.

The contribution of age structure to cell population responses to targeted therapeutics

Cells grown in culture act as a model system for analyzing the effects of anticancer compounds, which may affect cell behavior in a cell cycle position-dependent manner. Cell synchronization techniques have been generally employed to minimize the variation in cell cycle position. However, synchronization techniques are cumbersome and imprecise and the agents used to synchronize the cells potentially have other unknown effects on the cells. An alternative approach is to determine the age structure in the population and account for the cell cycle positional effects post hoc. Here we provide a formalism to use quantifiable lifespans from live cell microscopy experiments to parameterize an age-structured model of cell population response.