Virtual glioblastoma: growth, migration and treatment in a three-dimensional mathematical model

Objectives:  Glioblastomas are aggressive primary brain cancers that are characterized by extensive infiltration into the brain and are highly resistant to treatment. Through mathematical modelling, we model the process of invasion and predict the relative importance of mechanisms contributing to malignant invasion. Clinically, we predict patterns of tumour recurrence following various modes of therapeutic intervention.
Materials and methods:  Our mathematical model uses a realistic three-dimensional brain geometry and considers migrating and proliferating cells as separate classes. Several mechanisms for infiltrative migration are considered. Methods are developed for simulating surgical resection, radiotherapy and chemotherapy.
Results:  The model provides clinically realistic predictions of tumour growth and recurrence following therapeutic intervention. Specific results include (i) invasiveness is governed largely by the ability of glioblastoma cells to degrade and migrate through the extracellular matrix and the ability of single migrating cells to form colonies; (ii) tumours originating deeper in the brain generally grow more quickly than those of superficial origin; (iii) upon surgery, the margins and geometry of resection significantly determine the extent and pattern of postoperative recurrence; (iv) radiotherapy works synergistically with greater resection margins to reduce recurrence; (v) simulations in both two- and three-dimensional geometries give qualitatively similar results; and (vi) in an actual clinical case comprising several surgical interventions, the model provides good qualitative agreement between the simulated and observed course of the disease.
Conclusions:  The model provides a useful initial framework by which biological mechanisms of invasion and efficacy of potential treatment regimens may be assessed.     

Evolution of cell motility in an individual-based model of tumour growth

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.     

Effect of syndecan-1 overexpression on mesenchymal tumour cell proliferation with focus on different functional domains

Objectives:  Syndecan-1 is a transmembrane proteoglycan involved in various biological processes. Its extracellular, transmembrane and cytoplasmic domains may all participate in signal transduction. The aim of this study was to investigate the biological roles of these domains of syndecan-1.
Materials and methods:  We transfected cells of two mesenchymal tumour cell lines with a full-length syndecan-1 construct and three truncated variants, namely 78 construct lacking the EC domain with exception of DRKE sequence; 77 construct lacking extracellular the whole domain and RMKKK corresponding to a short cytoplasmic motif. Subcellular distribution was revealed using confocal laser microscopy. Overexpression of the constructs was verified using real-time RT-PCR and by FACS analysis and effects of syndecan-1 on cell behaviour were explored. Cell cycle analysis allowed for dissection of mechanisms regulating cell proliferation.
Results:  Overexpression of syndecan-1 influenced expression profile of the other syndecan members, and decreased tumour cell proliferation significantly by two mechanisms, as follows: increased length of G0/G1 phase was the most evident change in RMKKK and 77 transfectants, whereas prolonged S phase was more obvious in full-length transfectants. Overexpression of syndecan-1 changed the tumour cell morphology in an epithelioid direction.
Conclusions:  Both full-length and truncated syndecan-1 inhibited proliferation of the mesenchymal tumour cells, providing new insights into the importance for cancer growth of different functional domains of this proteoglycan.     

A Coupled Mathematical Model of Cell Migration,Vessel Cooption and Tumour Microenvironment during the Initiation of Micrometastases

We propose a coupled mathematical model for the detailed quantitative analyses of initial microtumour and micrometastases formation by including cancer cell migration, host vessel cooption and changes in microenvironment. Migrating cells are included as a new phenotype to describe the migration behaviour of malignant tumour cells. Migration probability of a migrating cell is assumed to be influenced by local chemical microenvironment. Pre-existing vessel cooption and remodelling are introduced according to the local haemodynamical microenvironment, such as interstitial pressure and vessel wall permeability. After the tumour cells and tumour vessels distribution are updated, the chemical substances are coupled calculated with the haemodynamical environment. The simulation results clearly reproduce the tumour cells migrate and proliferate along the pre-existing vessels at the very early stage of growth, which are consistent with many published experimental observations. In addition, the model demonstrates the interactions of tumour cells with the pre-existing vessels, which are believed to be essential for initial adhesion, proliferation, invasion, and micrometastases establishment. Quantitative analysis of tumour expansion in longitudinal and transverse directions shows that the cooption and migration along host vessels will be inhibited once angiogenesis phase occurs. The influences of the ability of cell migration and the inclusion of vessel cooption on the formation of micrometastases are discussed.     

The impact of phenotypic switching on glioblastoma growth and invasion

The brain tumour glioblastoma is characterised by diffuse and infiltrative growth into surrounding brain tissue. At the macroscopic level, the progression speed of a glioblastoma tumour is determined by two key factors: the cell proliferation rate and the cell migration speed. At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent in vivo imaging data. Here, we develop a mathematical model to analyse how the phenotypic switching between proliferative and migratory states of individual cells affects the macroscopic growth of the tumour. For this, we propose an individual-based stochastic model in which glioblastoma cells are either in a proliferative state, where they are stationary and divide, or in motile state in which they are subject to random motion. From the model we derive a continuum approximation in the form of two coupled reaction-diffusion equations, which exhibit travelling wave solutions whose speed of invasion depends on the model parameters. We propose a simple analytical method to predict progression rate from the cell-specific parameters and demonstrate that optimal glioblastoma growth depends on a non-trivial trade-off between the phenotypic switching rates. By linking cellular properties to an in vivo outcome, the model should be applicable to designing relevant cell screens for glioblastoma and cytometry-based patient prognostics.     

A color-coded orthotopic nude-mouse treatment model of brain-metastatic paralyzing spinal cord cancer that induces angiogenesis and neurogenesis

Objective: Cancer of the spinal cord is highly malignant and often leads to paralysis and death. A realistic mouse model would be an important benefit for the better understanding and treatment of spinal cord glioma.
Materials and methods: To develop an imageable, patient-like model of this disease, U87 human glioma tumour fragments (expressing red fluorescent protein), were transplanted by surgical orthotopic implantation into the spinal cord of nontransgenic nude mice or transgenic nude mice expressing nestin-driven green fluorescent protein (ND-GFP). In ND-GFP mice, GFP is expressed in nascent blood vessels and neural stem cells. The animals were treated with temozolomide or vehicle control.
Results: The intramedullary spinal cord tumour grew at the primary site, caused hind-limb paralysis and also metastasized to the brain. Temozolomide inhibited tumour growth ( P  < 0.01) and prevented metastasis, as well as prevented paralysis in four mice and delayed paralysis in two mice of the six tested ( P  = 0.005). In the ND-GFP-expressing host, ND-GFP cells staining positively for neuronal class III-β-tubulin or CD31, surrounded the tumour. These results suggest that the tumour stimulated both neurogenesis and angiogenesis, respectively.
Conclusion: A patient-like model of spinal cord glioma was thus developed, which can be used for the discovery of new agents, including those that inhibit invasion and metastasis of the disease as well as those that prevent paralysis.     

Using real-time impedance-based assays to monitor the effects of fibroblast-derived media on the adhesion,proliferation, migration and invasion of colon cancer cells

Increasing our knowledge of the mechanisms regulating cell proliferation, migration and invasion are central to understanding tumour progression and metastasis. The local tumour microenvironment contributes to the transformed phenotype in cancer by providing specific environmental cues that alter the cells behaviour and promotes metastasis. Fibroblasts have a strong association with cancer and in recent times there has been some emphasis in designing novel therapeutic strategies that alter fibroblast behaviour in the tumour microenvironment. Fibroblasts produce growth factors, chemokines and many of the proteins laid down in the ECM (extracellular matrix) that promote angiogenesis, inflammation and tumour progression. In this study, we use a label-free RTCA (real-time cell analysis) platform (xCELLigence) to investigate how media derived from human fibroblasts alters cancer cell behaviour. We used a series of complimentary and novel experimental approaches to show HCT116 cells adhere, proliferate and migrate significantly faster in the presence of media from human fibroblasts. As well as this, we used the xCELLigence CIM-plates system to show that HCT116 cells invade matrigel layers aggressively when migrating towards media derived from human fibroblasts. These data strongly suggest that fibroblasts have the ability to increase the migratory and invasive properties of HCT116 cells. This is the first study that provides real-time data on fibroblast-mediated migration and invasion kinetics of colon cancer cells.     

Phospholipase C-delta extends intercellular signalling range and responses to injury-released growth factors in non-excitable cells

Abstract.   Objectives : Intercellular communication in non-excitable cells is restricted to a limited range close to the signal source. Here, we have examined whether modification of the intracellular microenvironment could prolong the spatial proposition of signal generation and could increase cell proliferation . Material and methods : Mathematical models and experimental studies of endothelial repair after controlled mechanical injury were used. The models predict the diffusion range of injury-released growth factors and identify important parameters involved in a signalling regenerative mode. Transfected human umbilical vein endothelial cells (HUVECs) were used to validate model results, by examining intercellular calcium signalling range, cell proliferation and wound healing rate. Results : The models predict that growth factors have a limited capacity of extracellular diffusion and that intercellular signals are specially sensitive to cell phospholipase C-delta (PLCδ) levels. As basal PLCδ levels are increased by transfection, a significantly increased intercellular calcium range, enhanced cell proliferation, and faster wound healing rate were observed. Conclusion : Our in silico and in vitro studies demonstrated that non-excitable endothelial cells respond to stimuli in a complex manner, in which intercellular communication is controlled by physicochemical properties of the stimulus and by the cell microenvironment. Such findings may have profound implications for our understanding of the tight nature of autocrine cell growth control, compensation to stress states and response to altered microenvironment, under pathological conditions.     

Variation of broth composition by addition of broiler litter composting substrate extracts: influence on faecal bacterial growth

Aim:  This study investigates differences in bacterial growth response in broth amended with compost-substrate extracts periodically bypassed during broiler litter composting.
Methods and Results:  Compost samples, suspended in diluent were mixed with double strength broth into which ampicillin selective (0·3 g l⁻¹) Escherichia coli and E. faecalis were separately seeded. Growth was measured by viable cell count. The Levenberg–Marquardt algorithm was applied to obtain a four-parameter sigmoidal function that best described the diminishing height transitions of the curves for extracts of increasing composting age. The time course of the growth rate followed a unimodal bell-shaped curve. The Microfit^© application was run to generate information of direct microbiological interest: increasing λ and decreasing μ_max for both bacteria with time.
Conclusion:  More than the curve-fitting process, the Unified model option of the Microfit^© application has confirmed the significant differences ( P  <   0·05) in the growth curve behaviour with more stabilized substrate extracts. The study demonstrates further scopes for characterization of the sanitization potential and indirectly, the impact of indigenous microbial competitive exclusion effects on enteric bacteria.
Significance and Impact of the Study:  A different outlook to understanding faecal bacterial growth dynamics in compost has been presented, using predictive microbiology concepts. Further structured studies are needed to fine-tune the generality of the findings for model development.     

Collective dynamics of cancer cells confined in a confluent monolayer of normal cells

Tumorigenesis often involves specific changes in cell motility and intercellular adhesion. Understanding the collective cancer cell behavior associated with these specific changes could facilitate the detection of malignant characteristics during tumor growth and invasion. In this study, a cellular vertex model is developed to investigate the collective dynamics of a disk-like aggregate of cancer cells confined in a confluent monolayer of normal cells. The effects of intercellular adhesion and cell motility on tumor progression are examined. It is found that the stresses in both the cancer cells and the normal cells increase with tumor growth, resulting in a crowded environment and enhanced cell apoptosis. The intercellular adhesion between cancer cells and normal cells is revealed to promote tumor growth and invasion. The tumor invasion dynamics hinges on the motility of cancer cells. The cancer cells could orchestrate into different collective migration modes, e.g., directional migration and rotational oscillations, dictated by the competition between cell persistence and local coordination. Phase diagrams are established to reveal the competitive mechanisms. This work highlights the role of mechanics in regulating tumor growth and invasion.     

Gompertzian growth pattern correlated with phenotypic organization of colon carcinoma,malignant glioma and non-small cell lung carcinoma cell lines

Abstract. In the current study we present a Gompertzian model for cell growth as a function of cell phenotype using six human tumour cell lines (A-549, NCI-H596, NCI-H520, HT-29, SW-620 and U-251). Monolayer cells in exponential growth at various densities were quantified over a week by sulforhodamine B staining assay to produce cell-growth curves. A Gompertz equation was fitted to experimental data to obtain, for each cell line, three empirical growth parameters (initial cell density, cell-growth rate and carrying capacity – the maximal cell density). A cell-shape parameter named deformation coefficient D (a morphological relationship among spreading and confluent cells) was established and compared by regression analysis with the relative growth rate parameter K described by the Gompertz equation. We have found that coefficient D is directly proportional to the growth parameter K . The fit curve significantly matches the empirical data ( P  < 0.05), with a correlation coefficient of 0.9152. Therefore, a transformed Gompertzian growth function was obtained accordingly to D . The degree of correlation between the Gompertzian growth parameter and the coefficient D allows a new interpretation of the growth parameter K on the basis of morphological measurements of a set of tumour cell types, supporting the idea that cell-growth kinetics can be modulated by phenotypic organization of attached cells.     

Imaging of cell trafficking and metastases of paediatric rhabdomyosarcoma

Abstract.   Objective : The aim of this study was to establish a preclinical mouse model to study metastases of paediatric rhabdomyosarcoma at the macroscopic and cellular levels, with different imaging methods. Experimental Design : The alveolar rhabdomyosarcoma cell line Rh30 was stably transfected with the red fluorescent protein (DsRed2) then was xenotransplanted (intravenous injection [ n  = 8], and footpad injection [ n  = 8]) into nude mice (NMRI nu/nu). Macroscopic imaging of metastases was performed using DsRed2-fluorescence and flat-panel volumetric computed tomography scan. In a further series of animals ( n  = 8), in vivo cell trafficking of rhabdomyosarcoma cells using cellular imaging with an Olympus OV100 variable-magnification small-animal imaging system was used. Results : Metastases in the pelvis, thoracic wall and skin were visualized by fluorescence imaging. Pelvic metastases were found after tail vein injection and at other metastatic sites after footpad injection. Flat-panel volumetric computed tomography scan data allowed highly specific analysis of contrast between tumour and surrounding tissue. Correlation between fluorescence and flat-panel volumetric computed tomography scan imaging data was observed. Single-cell imaging visualized tumour cells in the vessels and demonstrated the arrest of tumour cells at vessel junctions followed by extravasation of the tumour cells. Conclusion : We established a model for visualization of experimental metastatic invasion and describe relevant tools for imaging childhood rhabdomyosarcoma metastases at the macroscopic and cellular levels. Imaging of cell trafficking visualized the behaviour of tumour cells and development of metastases by accumulation and extravasation of rhabdomyosarcoma cells.     

NPM1 Silencing Reduces Tumour Growth and MAPK Signalling in Prostate Cancer Cells

The chaperone nucleophosmin (NPM1) is over-expressed in the epithelial compartment of prostate tumours compared to adjacent healthy epithelium and may represent one of the key actors that support the neoplastic phenotype of prostate adenocarcinoma cells. Yet, the mechanisms that underlie NPM1 mediated phenotype remain elusive in the prostate. To better understand NPM1 functions in prostate cancer cells, we sought to characterize its impact on prostate cancer cells behaviour and decipher the mechanisms by which it may act. Here we show that NPM1 favors prostate tumour cell migration, invasion and colony forming. Furthermore, knockdown of NPM1 leads to a decrease in the growth of LNCaP-derived tumours grafted in Nude mice in vivo. Such oncogenic-like properties are found in conjunction with a positive regulation of NPM1 on the ERK1/2 (Extracellular signal-Regulated Kinases 1/2) kinase phosphorylation in response to EGF (Epidermal Growth Factor) stimulus, which is critical for prostate cancer progression following the setting of an autonomous production of the growth factor. NPM1 could then be a target to switch off specifically ERK1/2 pathway activation in order to decrease or inhibit cancer cell growth and migration.     

FGFs: crucial factors that regulate tumour initiation and progression

Fibroblast growth factors (FGFs) are crucial signalling molecules involved in normal cell growth, differentiation and proliferation. Over the past few decades, a large body of research has illustrated effects of individual FGFs on tumour initiation and progression. Tumour development is commonly accompanied with generation of new blood and lymph vessels, which support enhanced cell proliferation. Moreover, acquisition of tumour cells of the epithelial–mesenchymal transition (EMT) phenotype, enhances tumour cell migration and invasion potentials, crucial steps in tumour metastasis. This review summarizes recent findings concerning roles of FGFs in angiogenesis, lymphangiogenesis and EMT.     

Extracting information on the evolution of living- and dead-cell fractions of Salmonella Typhimurium colonies in gelatin gels based on microscopic images and plate-count data

Aims:  The aim of this study was to extract information on cell number and colony volume dynamics of Salmonella Typhimurium colonies.
Methods and Results:  Both cell number and colony volume of Salmonella Typhimurium in gelatin were monitored during the exponential and the stationary phase with varying pH and water activity, by plate counts and microscopic image analysis respectively. The exponential growth rates of cell numbers and colony volumes were correlated. The exponential growth rate of cell numbers was estimated based on this correlation and a secondary model that describes the effect of pH and water activity on the growth rate of the colony volumes. During the stationary phase, the cell number was constant, while colony volume increased, thus indicating the formation of a dead fraction. Models were developed to describe the living and dead population.
Conclusions:  By comparing colony volumes and cell numbers, the formation of dead fraction can be noticed from the beginning of the stationary phase, which indicates that the stationary phase is a dynamic – including both cell death and cell growth – rather than a static phase.
Significance and Impact of the Study:  This study was the first to investigate the proportion of living and dead bacteria within a stationary colony quantitatively.     

Travelling-wave analysis of a model of the immune response to cancer

In this paper we present a travelling-wave analysis of a mathematical model describing the growth of a solid tumour in the presence of an immune system response. From a modelling perspective, attention is focused upon the attack of tumour cells by tumour infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumour, without necrosis and at some stage prior to (tumour-induced) angiogenesis. As we have shown in previous work, for a particular choice of parameters, the underlying reaction-diffusion-chemotaxis system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy by depicting spatially heterogeneous tumour cell distributions that are characterized by a relatively small total number of tumour cells. This behaviour is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce bifurcations into the system, which in turn result in tumour invasion in the form of a standard travelling wave. The work presented in this paper complements the bifurcation analysis undertaken by Matzavinos et al. [Math. Med. Biol. IMA 21 (2004) 1-34] and establishes the existence of travelling-wave solutions for the system under discussion by promoting the understanding of the geometry of an appropriate phase space.     

Evolutionary game theory elucidates the role of glycolysis in glioma progression and invasion

Abstract. Objectives: Tumour progression has been described as a sequence of traits or phenotypes that cells have to acquire if the neoplasm is to become an invasive and malignant cancer. Although genetic mutations that lead to these phenotypes are random, the process by which some of these mutations become successful and cells spread is influenced by tumour microenvironment and the presence of other cell phenotypes. It is thus likely that some phenotypes that are essential in tumour progression will emerge in the tumour population only with prior presence of other different phenotypes. Materials and methods: In this study, we use evolutionary game theory to analyse the interactions between three different tumour cell phenotypes defined by autonomous growth, anaerobic glycolysis, and cancer cell invasion. The model allows us to understand certain specific aspects of glioma progression such as the emergence of diffuse tumour cell invasion in low‐grade tumours. Results: We have found that the invasive phenotype is more likely to evolve after appearance of the glycolytic phenotype which would explain the ubiquitous presence of invasive growth in malignant tumours. Conclusions: The result suggests that therapies, which increase the fitness cost of switching to anaerobic glycolysis, might decrease probability of the emergence of more invasive phenotypes.     

From passive diffusion to active cellular migration in mathematical models of tumour invasion

Mathematical models of tumour invasion appear as interesting tools for connecting the information extracted from medical imaging techniques and the large amount of data collected at the cellular and molecular levels. Most of the recent studies have used stochastic models of cell translocation for the comparison of computer simulations with histological solid tumour sections in order to discriminate and characterise expansive growth and active cell movements during host tissue invasion. This paper describes how a deterministic approach based on reaction-diffusion models and their generalisation in the mechano-chemical framework developed in the study of biological morphogenesis can be an alternative for analysing tumour morphological patterns. We support these considerations by reviewing two studies. In the first example, successful comparison of simulated brain tumour growth with a time sequence of computerised tomography (CT) scans leads to a quantification of the clinical parameters describing the invasion process and the therapy. The second example considers minimal hypotheses relating cell motility and cell traction forces. Using this model, we can simulate the bifurcation from an homogeneous distribution of cells at the tumour surface toward a nonhomogeneous density pattern which could characterise a pre-invasive stage at the tumour-host tissue interface.     

Density-dependent quiescence in glioma invasion: instability in a simple reaction-diffusion model for the migration/proliferation dichotomy

Gliomas are very aggressive brain tumours, in which tumour cells gain the ability to penetrate the surrounding normal tissue. The invasion mechanisms of this type of tumour remain to be elucidated. Our work is motivated by the migration/proliferation dichotomy (go-or-grow) hypothesis, i.e. the antagonistic migratory and proliferating cellular behaviours in a cell population, which may play a central role in these tumours. In this paper, we formulate a simple go-or-grow model to investigate the dynamics of a population of glioma cells for which the switch from a migratory to a proliferating phenotype (and vice versa) depends on the local cell density. The model consists of two reaction-diffusion equations describing cell migration, proliferation and a phenotypic switch. We use a combination of numerical and analytical techniques to characterize the development of spatio-temporal instabilities and travelling wave solutions generated by our model. We demonstrate that the density-dependent go-or-grow mechanism can produce complex dynamics similar to those associated with tumour heterogeneity and invasion.     

Quantifying the relative importance of niches and neutrality for coexistence in a model microbial system

1.  Ecologists have identified two types of processes promoting species coexistence: stabilizing mechanisms (niche differentiation and related processes) that increase negative intraspecific interactions relative to negative interspecific interactions, and equalizing mechanisms (neutrality) that minimize the differences in species' demographic parameters. It has been theoretically and empirically shown that the two types of mechanisms can operate simultaneously; however, their relative importance remains unstudied although this is a key question in the synthesis of niche and neutral theories.
2.  We experimentally quantified the relative importance of niche and neutral mechanisms in promoting phenotypic diversity in a model microbial system involving different phenotypes of the bacterium Pseudomonas fluorescens . Initially isogenic populations of the bacterium can diversify into a series of major and minor classes of phenotypes that can be treated as analogues of species. We estimated the relative population growth rate when rare of 32 phenotypes from six replicate microcosms. Each phenotype was assessed in a re-assembled microcosm in which the relative densities of all phenotypes remained the same except for the focal one which was reduced in frequency. A growth rate advantage when rare was considered evidence of non-neutral processes.
3.  Approximately one-third of the phenotypes had a growth rate advantage when rare while the remaining two-thirds showed neutral or near-neutral dynamics. Furthermore, there was overall little evidence that productivity increased with phenotypic diversity.
4.  Our results suggest that niche and neutral processes may simultaneously contribute to the maintenance of biodiversity, with the latter playing a more important role in our system, and that the operation of niche mechanisms does not necessarily lead to a positive biodiversity effect on ecosystem properties.