Interplay between distribution of live cells and growth dynamics of solid tumours

A mathematical model of wound healing in bovine corneal endothelium

We developed a mathematical model to describe healing processes in bovine corneal endothelial (BCE) cells in culture, triggered by mechanical wounds with parallel edges. Previous findings from our laboratory show that, in these cases, BCE monolayers exhibit an approximately constant healing velocity. Also, that caspase-dependent apoptosis occurs, with the fraction of apoptotic cells increasing with the distance traveled by the healing edge. In addition, in this study we report the novel findings that, for wound scratch assays performed preserving the basal extracellular matrix: i) the healing cells increase their en face surface area in a characteristic fashion, and ii) the average length of the segments of the cell columns actively participating in the healing process increases linearly with time. These latter observations preclude the utilization of standard traveling wave formalisms to model wound healing in BCE cells. Instead, we developed and studied a simple phenomenological model based on a plausible formula for the spreading dynamics of the individual healing cells, that incorporates original evidence about the process in BCE cells. The model can be simulated to: i) obtain an approximately constant healing velocity; ii) reproduce the profile of the healing cell areas, and iii) obtain approximately linear time dependences of the mean cell area and average length of the front active segments per column. In view of its accuracy to account for the experimental observations, the model can also be acceptably employed to quantify the appearance of apoptotic cells during BCE wound healing. The strategy utilized here could offer a novel formal framework to represent modifications undergone by some epithelial cell lines during wound healing.

The control of acidity in tumor cells: a biophysical model

Acidosis of the tumor microenvironment leads to cancer invasion, progression and resistance to therapies. We present a biophysical model that describes how tumor cells regulate intracellular and extracellular acidity while they grow in a microenvironment characterized by increasing acidity and hypoxia. The model takes into account the dynamic interplay between glucose and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {O}_2$$\end{document}O2 consumption with lactate and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CO}_2$$\end{document}CO2 production and connects these processes to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}^+$$\end{document}H+ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {HCO}_3^-$$\end{document}HCO3- fluxes inside and outside cells. We have validated the model with independent experimental data and used it to investigate how and to which extent tumor cells can survive in adverse micro-environments characterized by acidity and hypoxia. The simulations show a dominance of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}^+$$\end{document}H+ exchanges in well-oxygenated regions, and of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {HCO}_3^-$$\end{document}HCO3- exchanges in the inner hypoxic regions where tumor cells are known to acquire malignant phenotypes. The model also includes the activity of the enzyme Carbonic Anhydrase 9 (CA9), a known marker of tumor aggressiveness, and the simulations demonstrate that CA9 acts as a nonlinear \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {pH}_i$$\end{document}pHi equalizer at any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {O}_2$$\end{document}O2 level in cells that grow in acidic extracellular environments.

X-ray microtomography as a new approach for imaging and analysis of tumor spheroids

Three-dimensional (3D) spheroids mimic important properties of tumors and may soon become a reasonable substitute for animal models and human tissue, eliminating numerous problems related to in vivo and ex vivo experiments and pre-clinical drug trials. Currently, various imaging methods including X-ray microtomography (micro-CT), exist but their spatial resolution is limited. Here, we visualized and provided a morphological analysis of spheroid cell cultures using micro-CT and compared it to that of confocal microscopy. An approach is proposed that can potentially open new diagnostic opportunities to determine the morphology of cancer cells cultured in 3D structures instead of using actual tumors. Spheroids were formed from human melanoma cell lines WM266-4 and WM115 seeded at different cell densities using the hanging drop method. Micro-CT analysis of spheroid showed that spheroid size and shape differed depending on the cell line, initial cell number, and duration of culture. The melanoma cell lines used in this study can successfully be cultured as 3D spheroids and used to substitute human and animal models in pre-clinical studies. The micro-CT allows for high-resolution visualization of the spheroids structure.

Oxygen in the Tumor Microenvironment: Mathematical and Numerical Modeling

There are many reasons to try to achieve a good grasp of the distribution of oxygen in the tumor microenvironment. The lack of oxygen - hypoxia - is a main actor in the evolution of tumors and in their growth and appears to be just as important in tumor invasion and metastasis. Mathematical models of the distribution of oxygen in tumors which are based on reaction-diffusion equations provide partial but qualitatively significant descriptions of the measured oxygen concentrations in the tumor microenvironment, especially when they incorporate important elements of the blood vessel network such as the blood vessel size and spatial distribution and the pulsation of local pressure due to blood circulation. Here, we review our mathematical and numerical approaches to the distribution of oxygen that yield insights both on the role of the distribution of blood vessel density and size and on the fluctuations of blood pressure.

Fine-grained simulations of the microenvironment of vascularized tumours

One of many important features of the tumour microenvironment is that it is a place of active Darwinian selection where different tumour clones become adapted to the variety of ecological niches that make up the microenvironment. These evolutionary processes turn the microenvironment into a powerful source of tumour heterogeneity and contribute to the development of drug resistance in cancer. Here, we describe a computational tool to study the ecology of the microenvironment and report results about the ecology of the tumour microenvironment and its evolutionary dynamics.

Collective radioresistance of T47D breast carcinoma cells is mediated by a Syncytin-1 homologous protein

It is generally accepted that radiotherapy must target clonogenic cells, i.e., those cells in a tumour that have self-renewing potential. Focussing on isolated clonogenic cells, however, may lead to an underestimate or even to an outright neglect of the importance of biological mechanisms that regulate tumour cell sensitivity to radiation. We develop a new statistical and experimental approach to quantify the effects of radiation on cell populations as a whole. In our experiments, we change the proximity relationships of the cells by culturing them in wells with different shapes, and we find that the radiosensitivity of T47D human breast carcinoma cells in tight clusters is different from that of isolated cells. Molecular analyses show that T47D cells express a Syncytin-1 homologous protein (SyHP). We observe that SyHP translocates to the external surface of the plasma membrane of cells killed by radiation treatment. The data support the fundamental role of SyHP in the formation of intercellular cytoplasmic bridges and in the enhanced radioresistance of surviving cells. We conclude that complex and unexpected biological mechanisms of tumour radioresistance take place at the cell population level. These mechanisms may significantly bias our estimates of the radiosensitivity of breast carcinomas in vivo and thereby affect treatment plans, and they call for further investigations.

Improving the Subcutaneous Mouse Tumor Model by Effective Manipulation of Magnetic Nanoparticles-Treated Implanted Cancer Cells

Murine tumor models have played a fundamental role in the development of novel therapeutic interventions and are currently widely used in translational research. Specifically, strategies that aim at reducing inter-animal variability of tumor size in transplantable mouse tumor models are of particular importance. In our approach, we used magnetic nanoparticles to label and manipulate colon cancer cells for the improvement of the standard syngeneic subcutaneous mouse tumor model. Following subcutaneous injection on the scruff of the neck, magnetically-tagged implanted cancer cells were manipulated by applying an external magnetic field towards localized tumor formation. Our data provide evidence that this approach can facilitate the formation of localized tumors of similar shape, reducing thereby the tumor size's variability. For validating the proof-of-principle, a low-dose of 5-FU was administered in small animal groups as a representative anticancer therapy. Under these experimental conditions, the 5-FU-induced tumor growth inhibition was statistically significant only after the implementation of the proposed method. The presented approach is a promising strategy for studying accurately therapeutic interventions in subcutaneous experimental solid tumor models allowing for the detection of statistically significant differences between smaller experimental groups.

Allometric scaling in-vitro

About two decades ago, West and coworkers established a model which predicts that metabolic rate follows a three quarter power relationship with the mass of an organism, based on the premise that tissues are supplied nutrients through a fractal distribution network. Quarter power scaling is widely considered a universal law of biology and it is generally accepted that were in-vitro cultures to obey allometric metabolic scaling, they would have more predictive potential and could, for instance, provide a viable substitute for animals in research. This paper outlines a theoretical and computational framework for establishing quarter power scaling in three-dimensional spherical constructs in-vitro, starting where fractal distribution ends. Allometric scaling in non-vascular spherical tissue constructs was assessed using models of Michaelis Menten oxygen consumption and diffusion. The models demonstrate that physiological scaling is maintained when about 5 to 60% of the construct is exposed to oxygen concentrations less than the Michaelis Menten constant, with a significant concentration gradient in the sphere. The results have important implications for the design of downscaled in-vitro systems with physiological relevance.

Pulsation-limited oxygen diffusion in the tumour microenvironment

Hypoxia is central to tumour evolution, growth, invasion and metastasis. Mathematical models of hypoxia based on reaction-diffusion equations provide seemingly incomplete descriptions as they fail to predict the measured oxygen concentrations in the tumour microenvironment. In an attempt to explain the discrepancies, we consider both the inhomogeneous distribution of oxygen-consuming cells in solid tumours and the dynamics of blood flow in the tumour microcirculation. We find that the low-frequency oscillations play an important role in the establishment of tumour hypoxia. The oscillations interact with consumption to inhibit oxygen diffusion in the microenvironment. This suggests that alpha-blockers-a class of drugs used to treat hypertension and stress disorders, and known to lower or even abolish low-frequency oscillations of arterial blood flow -may act as adjuvant drugs in the radiotherapy of solid tumours by enhancing the oxygen effect.

Echo Chambers: Emotional Contagion and Group Polarization on Facebook

Recent findings showed that users on Facebook tend to select information that adhere to their system of beliefs and to form polarized groups - i.e., echo chambers. Such a tendency dominates information cascades and might affect public debates on social relevant issues. In this work we explore the structural evolution of communities of interest by accounting for users emotions and engagement. Focusing on the Facebook pages reporting on scientific and conspiracy content, we characterize the evolution of the size of the two communities by fitting daily resolution data with three growth models - i.e. the Gompertz model, the Logistic model, and the Log-logistic model. Although all the models appropriately describe the data structure, the Logistic one shows the best fit. Then, we explore the interplay between emotional state and engagement of users in the group dynamics. Our findings show that communities' emotional behavior is affected by the users' involvement inside the echo chamber. Indeed, to an higher involvement corresponds a more negative approach. Moreover, we observe that, on average, more active users show a faster shift towards the negativity than less active ones.

Metabolic scaling in solid tumours

Tumour metabolism is an outstanding topic of cancer research, as it determines the growth rate and the global activity of tumours. Recently, by combining the diffusion of oxygen, nutrients, and metabolites in the extracellular environment, and the internal motions that mix live and dead cells, we derived a growth law of solid tumours which is linked to parameters at the cellular level1. Here we use this growth law to obtain a metabolic scaling law for solid tumours, which is obeyed by tumours of different histotypes both in vitro and in vivo, and we display its relation with the fractal dimension of the distribution of live cells in the tumour mass. The scaling behaviour is related to measurable parameters, with potential applications in the clinical practice.

BGFit: management and automated fitting of biological growth curves

BACKGROUND

Existing tools to model cell growth curves do not offer a flexible integrative approach to manage large datasets and automatically estimate parameters. Due to the increase of experimental time-series from microbiology and oncology, the need for a software that allows researchers to easily organize experimental data and simultaneously extract relevant parameters in an efficient way is crucial.

RESULTS

BGFit provides a web-based unified platform, where a rich set of dynamic models can be fitted to experimental time-series data, further allowing to efficiently manage the results in a structured and hierarchical way. The data managing system allows to organize projects, experiments and measurements data and also to define teams with different editing and viewing permission. Several dynamic and algebraic models are already implemented, such as polynomial regression, Gompertz, Baranyi, Logistic and Live Cell Fraction models and the user can add easily new models thus expanding current ones.

CONCLUSIONS

BGFit allows users to easily manage their data and models in an integrated way, even if they are not familiar with databases or existing computational tools for parameter estimation. BGFit is designed with a flexible architecture that focus on extensibility and leverages free software with existing tools and methods, allowing to compare and evaluate different data modeling techniques. The application is described in the context of bacterial and tumor cells growth data fitting, but it is also applicable to any type of two-dimensional data, e.g. physical chemistry and macroeconomic time series, being fully scalable to high number of projects, data and model complexity.

Computer-aided biophysical modeling: a quantitative approach to complex biological systems

When dealing with the biophysics of tumors, analytical and numerical modeling tools have long been regarded as potentially useful but practically immature tools. Further developments could not just overturn this predicament, but lead to completely new perspectives in biology. Here, we give an account of our own computational tool and how we have put it to good use, and we discuss a paradigmatic example to outline a path to making cell biology more quantitative and predictive.