Multicellular tumor spheroid in an off-lattice Voronoi-Delaunay cell model

3D Hydrogel Encapsulation Regulates Nephrogenesis in Kidney Organoids

Stem cell-derived kidney organoids contain nephron segments that recapitulate morphological and functional aspects of the human kidney. However, directed differentiation protocols for kidney organoids are largely conducted using biochemical signals to control differentiation. Here, the hypothesis that mechanical signals regulate nephrogenesis is investigated in 3D culture by encapsulating kidney organoids within viscoelastic alginate hydrogels with varying rates of stress relaxation. Tubular nephron segments are significantly more convoluted in kidney organoids differentiated in encapsulating hydrogels when compared with those in suspension culture. Hydrogel viscoelasticity regulates the spatial distribution of nephron segments within the differentiating kidney organoids. Consistent with these observations, a particle-based computational model predicts that the extent of deformation of the hydrogel-organoid interface regulates the morphology of nephron segments. Elevated extracellular calcium levels in the culture medium, which can be impacted by the hydrogels, decrease the glomerulus-to-tubule ratio of nephron segments. These findings reveal that hydrogel encapsulation regulates nephron patterning and morphology and suggest that the mechanical microenvironment is an important design variable for kidney regenerative medicine.

Calibration of agent based models for monophasic and biphasic tumour growth using approximate Bayesian computation

Agent-based models (ABMs) are readily used to capture the stochasticity in tumour evolution; however, these models are often challenging to validate with experimental measurements due to model complexity. The Voronoi cell-based model (VCBM) is an off-lattice agent-based model that captures individual cell shapes using a Voronoi tessellation and mimics the evolution of cancer cell proliferation and movement. Evidence suggests tumours can exhibit biphasic growth in vivo. To account for this phenomena, we extend the VCBM to capture the existence of two distinct growth phases. Prior work primarily focused on point estimation for the parameters without consideration of estimating uncertainty. In this paper, approximate Bayesian computation is employed to calibrate the model to in vivo measurements of breast, ovarian and pancreatic cancer. Our approach involves estimating the distribution of parameters that govern cancer cell proliferation and recovering outputs that match the experimental data. Our results show that the VCBM, and its biphasic extension, provides insight into tumour growth and quantifies uncertainty in the switching time between the two phases of the biphasic growth model. We find this approach enables precise estimates for the time taken for a daughter cell to become a mature cell. This allows us to propose future refinements to the model to improve accuracy, whilst also making conclusions about the differences in cancer cell characteristics.

Free volume theory explains the unusual behavior of viscosity in a non-confluent tissue during morphogenesis

A recent experiment on zebrafish blastoderm morphogenesis showed that the viscosity (η) of a non-confluent embryonic tissue grows sharply until a critical cell packing fraction (ϕS). The increase in η up to ϕS is similar to the behavior observed in several glass-forming materials, which suggests that the cell dynamics is sluggish or glass-like. Surprisingly, η is a constant above ϕS. To determine the mechanism of this unusual dependence of η on ϕ, we performed extensive simulations using an agent-based model of a dense non-confluent two-dimensional tissue. We show that polydispersity in the cell size, and the propensity of the cells to deform, results in the saturation of the available free area per cell beyond a critical packing fraction. Saturation in the free space not only explains the viscosity plateau above ϕS but also provides a relationship between equilibrium geometrical packing to the dramatic increase in the relaxation dynamics.

Bridging the gap between collective motility and epithelial-mesenchymal transitions through the active finite voronoi model

We introduce an active version of the recently proposed finite Voronoi model of epithelial tissue. The resultant Active Finite Voronoi (AFV) model enables the study of both confluent and non-confluent geometries and transitions between them, in the presence of active cells. Our study identifies six distinct phases, characterized by aggregation-segregation, dynamical jamming-unjamming, and epithelial-mesenchymal transitions (EMT), thereby extending the behavior beyond that observed in previously studied vertex-based models. The AFV model with rich phase diagram provides a cohesive framework that unifies the well-observed progression to collective motility via unjamming with the intricate dynamics enabled by EMT. This approach should prove useful for challenges in developmental biology systems as well as the complex context of cancer metastasis. The simulation code is also provided.

Free and Interfacial Boundaries in Individual-Based Models of Multicellular Biological systems

Coordination of cell behaviour is key to a myriad of biological processes including tissue morphogenesis, wound healing, and tumour growth. As such, individual-based computational models, which explicitly describe inter-cellular interactions, are commonly used to model collective cell dynamics. However, when using individual-based models, it is unclear how descriptions of cell boundaries affect overall population dynamics. In order to investigate this we define three cell boundary descriptions of varying complexities for each of three widely used off-lattice individual-based models: overlapping spheres, Voronoi tessellation, and vertex models. We apply our models to multiple biological scenarios to investigate how cell boundary description can influence tissue-scale behaviour. We find that the Voronoi tessellation model is most sensitive to changes in the cell boundary description with basic models being inappropriate in many cases. The timescale of tissue evolution when using an overlapping spheres model is coupled to the boundary description. The vertex model is demonstrated to be the most stable to changes in boundary description, though still exhibits timescale sensitivity. When using individual-based computational models one should carefully consider how cell boundaries are defined. To inform future work, we provide an exploration of common individual-based models and cell boundary descriptions in frequently studied biological scenarios and discuss their benefits and disadvantages.

Enhanced mechanical heterogeneity of cell collectives due to temporal fluctuations in cell elasticity

Cells are dynamic systems characterized by temporal variations in biophysical properties such as stiffness and contractility. Recent studies show that the recruitment and release of actin filaments into and out of the cell cortex-a network of proteins underneath the cell membrane-leads to cell stiffening prior to division and softening immediately afterward. In three-dimensional (3D) cell collectives, it is unclear whether the stiffness change during division at the single-cell scale controls the spatial structure and dynamics at the multicellular scale. This is an important question to understand because cell stiffness variations impact cell spatial organization and cancer progression. Using a minimal 3D model incorporating cell birth, death, and cell-to-cell elastic and adhesive interactions, we investigate the effect of mechanical heterogeneity-variations in individual cell stiffnesses that make up the cell collective-on tumor spatial organization and cell dynamics. We discover that spatial mechanical heterogeneity characterized by a spheroid core composed of stiffer cells and softer cells in the periphery emerges within dense 3D cell collectives, which may be a general feature of multicellular tumor growth. We show that heightened spatial mechanical heterogeneity enhances single-cell dynamics and volumetric tumor growth driven by fluctuations in cell elasticity. Our results could have important implications in understanding how spatiotemporal variations in single-cell stiffness determine tumor growth and spread.

Adhesion strength between cells regulate nonmonotonic growth by a biomechanical feedback mechanism

We determine how intercellular interactions and mechanical pressure experienced by single cells regulate cell proliferation using a minimal computational model for three-dimensional multicellular spheroid (MCS) growth. We discover that emergent spatial variations in the cell division rate, depending on the location of the cells either at the core or periphery within the MCS, is regulated by intercellular adhesion strength (fad). Varying fad results in nonmonotonic proliferation of cells in the MCS. A biomechanical feedback mechanism coupling the fad and microenvironment-dependent pressure fluctuations relative to a threshold value (pc) determines the onset of a dormant phase, and explains the nonmonotonic proliferation response. Increasing fad from low values enhances cell proliferation because pressure on individual cells is smaller compared with pc. However, at high fad, cells readily become dormant and cannot rearrange effectively in spacetime, leading to arrested cell proliferation. Utilizing our theoretical predictions, we explain experimental data on the impact of adhesion strength on cell proliferation and find good agreement. Our work, which shows that proliferation is regulated by pressure-adhesion feedback mechanism, may be a general feature of multicellular growth.

Mechanical feedback controls the emergence of dynamical memory in growing tissue monolayers

The growth of a tissue, which depends on cell–cell interactions and biologically relevant processes such as cell division and apoptosis, is regulated by a mechanical feedback mechanism. We account for these effects in a minimal two-dimensional model in order to investigate the consequences of mechanical feedback, which is controlled by a critical pressure, p c. A cell can only grow and divide if its pressure, due to interaction with its neighbors, is less than p c. Because temperature is not a relevant variable, the cell dynamics is driven by self-generated active forces (SGAFs) that arise due to cell division. We show that even in the absence of intercellular interactions, cells undergo diffusive behavior. The SGAF-driven diffusion is indistinguishable from the well-known dynamics of a free Brownian particle at a fixed finite temperature. When intercellular interactions are taken into account, we find persistent temporal correlations in the force–force autocorrelation function (FAF) that extends over a timescale of several cell division times. The time-dependence of the FAF reveals memory effects, which increases as p c increases. The observed non-Markovian effects emerge due to the interplay of cell division and mechanical feedback and are inherently a non-equilibrium phenomenon.

Agent-based computational modeling of glioblastoma predicts that stromal density is central to oncolytic virus efficacy

Oncolytic viruses (OVs) are emerging cancer immunotherapy. Despite notable successes in the treatment of some tumors, OV therapy for central nervous system cancers has failed to show efficacy. We used an ex vivo tumor model developed from human glioblastoma tissue to evaluate the infiltration of herpes simplex OV rQNestin (oHSV-1) into glioblastoma tumors. We next leveraged our data to develop a computational, model of glioblastoma dynamics that accounts for cellular interactions within the tumor. Using our computational model, we found that low stromal density was highly predictive of oHSV-1 therapeutic success, suggesting that the efficacy of oHSV-1 in glioblastoma may be determined by stromal-to-tumor cell regional density. We validated these findings in heterogenous patient samples from brain metastatic adenocarcinoma. Our integrated modeling strategy can be applied to suggest mechanisms of therapeutic responses for central nervous system cancers and to facilitate the successful translation of OVs into the clinic.

Quantitative modeling identifies critical cell mechanics driving bile duct lumen formation

Biliary ducts collect bile from liver lobules, the smallest functional and anatomical units of liver, and carry it to the gallbladder. Disruptions in this process caused by defective embryonic development, or through ductal reaction in liver disease have a major impact on life quality and survival of patients. A deep understanding of the processes underlying bile duct lumen formation is crucial to identify intervention points to avoid or treat the appearance of defective bile ducts. Several hypotheses have been proposed to characterize the biophysical mechanisms driving initial bile duct lumen formation during embryogenesis. Here, guided by the quantification of morphological features and expression of genes in bile ducts from embryonic mouse liver, we sharpened these hypotheses and collected data to develop a high resolution individual cell-based computational model that enables to test alternative hypotheses in silico. This model permits realistic simulations of tissue and cell mechanics at sub-cellular scale. Our simulations suggest that successful bile duct lumen formation requires a simultaneous contribution of directed cell division of cholangiocytes, local osmotic effects generated by salt excretion in the lumen, and temporally-controlled differentiation of hepatoblasts to cholangiocytes, with apical constriction of cholangiocytes only moderately affecting luminal size.

The initial step in bile duct development is the formation of a biliary lumen, a process which involves several cellular mechanisms, such as cell division and polarization, and secretion of fluid. However, how these mechanisms are orchestrated in time and space is difficult to understand. Here, we built a computational model of biliary lumen formation which represents every cell and its function in detail. With the model we can simulate the effect of biophysical aspects that affect duct formation. We have tested the individual and combined effects of directed cell division, apical constriction, and osmotic effects on lumen expansion by varying the parameters that control their relative strength. Our simulations suggest that successful bile duct lumen formation requires the simultaneous contribution of directed cell division of cholangiocytes, local osmotic effects generated by salt excretion in the lumen, and temporally-controlled differentiation of hepatoblasts to cholangiocytes, with apical constriction of cholangiocytes only moderately affecting luminal size.

CBMOS: a GPU-enabled Python framework for the numerical study of center-based models

BACKGROUND

Cell-based models are becoming increasingly popular for applications in developmental biology. However, the impact of numerical choices on the accuracy and efficiency of the simulation of these models is rarely meticulously tested. Without concrete studies to differentiate between solid model conclusions and numerical artifacts, modelers are at risk of being misled by their experiments' results. Most cell-based modeling frameworks offer a feature-rich environment, providing a wide range of biological components, but are less suitable for numerical studies. There is thus a need for software specifically targeted at this use case.

RESULTS

We present CBMOS, a Python framework for the simulation of the center-based or cell-centered model. Contrary to other implementations, CBMOS' focus is on facilitating numerical study of center-based models by providing access to multiple ordinary differential equation solvers and force functions through a flexible, user-friendly interface and by enabling rapid testing through graphics processing unit (GPU) acceleration. We show-case its potential by illustrating two common workflows: (1) comparison of the numerical properties of two solvers within a Jupyter notebook and (2) measuring average wall times of both solvers on a high performance computing cluster. More specifically, we confirm that although for moderate accuracy levels the backward Euler method allows for larger time step sizes than the commonly used forward Euler method, its additional computational cost due to being an implicit method prohibits its use for practical test cases.

CONCLUSIONS

CBMOS is a flexible, easy-to-use Python implementation of the center-based model, exposing both basic model assumptions and numerical components to the user. It is available on GitHub and PyPI under an MIT license. CBMOS allows for fast prototyping on a central processing unit for small systems through the use of NumPy. Using CuPy on a GPU, cell populations of up to 10,000 cells can be simulated within a few seconds. As such, it will substantially lower the time investment for any modeler to check the crucial assumption that model conclusions are independent of numerical issues.

Derivation of continuum models from discrete models of mechanical forces in cell populations

In certain discrete models of populations of biological cells, the mechanical forces between the cells are center based or vertex based on the microscopic level where each cell is individually represented. The cells are circular or spherical in a center based model and polygonal or polyhedral in a vertex based model. On a higher, macroscopic level, the time evolution of the density of the cells is described by partial differential equations (PDEs). We derive relations between the modelling on the micro and macro levels in one, two, and three dimensions by regarding the micro model as a discretization of a PDE for conservation of mass on the macro level. The forces in the micro model correspond on the macro level to a gradient of the pressure scaled by quantities depending on the cell geometry. The two levels of modelling are compared in numerical experiments in one and two dimensions.

Interparticle Adhesion Regulates the Surface Roughness of Growing Dense Three-Dimensional Active Particle Aggregates

Activity and self-generated motion are fundamental features observed in many living and nonliving systems. Given that interparticle adhesive forces can regulate particle dynamics, we investigate how interparticle adhesion strength controls the boundary growth and roughness of active particle aggregates. Using particle based simulations incorporating both activity (birth, death, and growth) and systematic physical interactions (elasticity and adhesion), we establish that interparticle adhesion strength (f^ad) controls the surface roughness of a densely packed three-dimensional(3D) active particle aggregate expanding into a highly viscous medium. We discover that the surface roughness of a 3D active particle aggregate increases in proportion to the interparticle adhesion strength (f^ad) and show that asymmetry in the radial and transverse active particle mean-squared displacement (MSD) suppresses 3D surface roughness at lower adhesion strengths. By analyzing the statistical properties of particle displacements at the aggregate periphery, we determine that the 3D surface roughness is driven by the movement of active particle toward the core at high interparticle adhesion strengths. Our results elucidate the physics controlling the expansion of adhesive 3D active particle collectives into a highly viscous medium, with implications into understanding stochastic interface growth in active matter systems characterized by self-generation of particles.

CPOP: An open source C++ cell POPulation modeler for radiation biology applications

PURPOSE

Multicellular tumor spheroids are realistic in-vitro systems used in radiation biology research to study the effect of anticancer drugs or to evaluate the resistance of cancer cells under specific conditions. When combining the modeling of spheroids together with the simulation of radiation using Monte Carlo methods, one could estimate cell and DNA damage to be compared with experimental data. We developed a Cell Population (CPOP) modeler combined to Geant4 simulations in order to tackle how energy depositions are allocated to cells, especially when enhancing radiation outcomes using high-Z nanoparticles. CPOP manages to model large three-dimensional cell populations with independent deformable cells described with their nucleus, cytoplasm and membranes together with force law systems to manage cell-cell interactions.

METHODS

CPOP is an opensource platform written in C++. It is divided into two main libraries: a "Modeler" library, for cell geometry modeling using meshes, and a Multi Agent System (MAS) library, simulating all agent (cell) interactions among the population. CPOP is fully interfaced with the Geant4 Monte Carlo toolkit and is able to directly launch Geant4 simulations after compilation. We modeled a full and realistic 3D cell population from SK-MEL28 melanoma cell population cultured experimentally. The spheroid diameter of 550 ± 40 µm corresponds to a population of approximately 1000 cells having a diameter of 17.2 ± 2.5 µm and a nucleus diameter of 11.2 ± 2.0 µm. We decided to reproduce cell irradiations performed with a X-RAD 320 Biological Irradiator (Precision XRay Inc., North Branford, CT).

RESULTS

We simulated the energy spectrum of secondary particles generated in the vicinity of the spheroid and plotted the different energy spectra recovered internally to the spheroid. We evaluated also the impact of AGuIX (Gadolinium) nanoparticles modeled into the spheroid with their corresponding secondary energy spectra.

CONCLUSIONS

We succeeded into modeling cell populations and combined them with Geant4 simulations. The next step will be to integrate DNA geometrical models into cell nuclei and to use the Geant4-DNA physics and radiolysis modeling capabilities in order to evaluate early strand breaks induced on DNA.

Fluctuation-Induced Phase Separation in Metric and Topological Models of Collective Motion

We study the role of noise on the nature of the transition to collective motion in dry active matter. Starting from field theories that predict a continuous transition at the deterministic level, we show that fluctuations induce a density-dependent shift of the onset of order, which in turn changes the nature of the transition into a phase-separation scenario. Our results apply to a range of systems, including models in which particles interact with their "topological" neighbors that have been believed so far to exhibit a continuous onset of order. Our analytical predictions are confirmed by numerical simulations of fluctuating hydrodynamics and microscopic models.

Vangl2 promotes the formation of long cytonemes to enable distant Wnt/β-catenin signaling

Wnt signaling regulates cell proliferation and cell differentiation as well as migration and polarity during development. However, it is still unclear how the Wnt ligand distribution is precisely controlled to fulfil these functions. Here, we show that the planar cell polarity protein Vangl2 regulates the distribution of Wnt by cytonemes. In zebrafish epiblast cells, mouse intestinal telocytes and human gastric cancer cells, Vangl2 activation generates extremely long cytonemes, which branch and deliver Wnt protein to multiple cells. The Vangl2-activated cytonemes increase Wnt/β-catenin signaling in the surrounding cells. Concordantly, Vangl2 inhibition causes fewer and shorter cytonemes to be formed and reduces paracrine Wnt/β-catenin signaling. A mathematical model simulating these Vangl2 functions on cytonemes in zebrafish gastrulation predicts a shift of the signaling gradient, altered tissue patterning, and a loss of tissue domain sharpness. We confirmed these predictions during anteroposterior patterning in the zebrafish neural plate. In summary, we demonstrate that Vangl2 is fundamental to paracrine Wnt/β-catenin signaling by controlling cytoneme behaviour.

Cell – extracellular matrix interaction in glioma growth. In silico model

The study aims to investigate the role of viscoelastic interactions between cells and extracellular matrix (ECM) in avascular tumor growth. Computer simulations of glioma multicellular tumor spheroid (MTS) growth are being carried out for various conditions. The calculations are based on a continuous model, which simulates oxygen transport into MTS; transitions between three cell phenotypes, cell transport, conditioned by hydrostatic forces in cell–ECM composite system, cell motility and cell adhesion. Visco-elastic cell aggregation and elastic ECM scaffold represent two compressible constituents of the composite. Cell–ECM interactions form a Transition Layer on the spheroid surface, where mechanical characteristics of tumor undergo rapid transition. This layer facilitates tumor progression to a great extent. The study demonstrates strong effects of ECM stiffness, mechanical deformations of the matrix and cell–cell adhesion on tumor progression. The simulations show in particular that at certain, rather high degrees of matrix stiffness a formation of distant multicellular clusters takes place, while at further increase of ECM stiffness subtumors do not form. The model also illustrates to what extent mere mechanical properties of cell–ECM system may contribute into variations of glioma invasion scenarios.

Tracing the footsteps of autophagy in computational biology

Autophagy plays a crucial role in maintaining cellular homeostasis through the degradation of unwanted materials like damaged mitochondria and misfolded proteins. However, the contribution of autophagy toward a healthy cell environment is not only limited to the cleaning process. It also assists in protein synthesis when the system lacks the amino acids’ inflow from the extracellular environment due to diet consumptions. Reduction in the autophagy process is associated with diseases like cancer, diabetes, non-alcoholic steatohepatitis, etc., while uncontrolled autophagy may facilitate cell death. We need a better understanding of the autophagy processes and their regulatory mechanisms at various levels (molecules, cells, tissues). This demands a thorough understanding of the system with the help of mathematical and computational tools. The present review illuminates how systems biology approaches are being used for the study of the autophagy process. A comprehensive insight is provided on the application of computational methods involving mathematical modeling and network analysis in the autophagy process. Various mathematical models based on the system of differential equations for studying autophagy are covered here. We have also highlighted the significance of network analysis and machine learning in capturing the core regulatory machinery governing the autophagy process. We explored the available autophagic databases and related resources along with their attributes that are useful in investigating autophagy through computational methods. We conclude the article addressing the potential future perspective in this area, which might provide a more in-depth insight into the dynamics of autophagy.

Impact of Force Function Formulations on the Numerical Simulation of Centre-Based Models

Centre-based or cell-centre models are a framework for the computational study of multicellular systems with widespread use in cancer modelling and computational developmental biology. At the core of these models are the numerical method used to update cell positions and the force functions that encode the pairwise mechanical interactions of cells. For the latter, there are multiple choices that could potentially affect both the biological behaviour captured, and the robustness and efficiency of simulation. For example, available open-source software implementations of centre-based models rely on different force functions for their default behaviour and it is not straightforward for a modeller to know if these are interchangeable. Our study addresses this problem and contributes to the understanding of the potential and limitations of three popular force functions from a numerical perspective. We show empirically that choosing the force parameters such that the relaxation time for two cells after cell division is consistent between different force functions results in good agreement of the population radius of a two-dimensional monolayer relaxing mechanically after intense cell proliferation. Furthermore, we report that numerical stability is not sufficient to prevent unphysical cell trajectories following cell division, and consequently, that too large time steps can cause geometrical differences at the population level.

Mathematical models for cell migration: a non-local perspective

We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell-cell and cell-tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Investigating the Physical Effects in Bacterial Therapies for Avascular Tumors

Tumor-targeting bacteria elicit anticancer effects by infiltrating hypoxic regions, releasing toxic agents and inducing immune responses. Although current research has largely focused on the influence of chemical and immunological aspects on the mechanisms of bacterial therapy, the impact of physical effects is still elusive. Here, we propose a mathematical model for the anti-tumor activity of bacteria in avascular tumors that takes into account the relevant chemo-mechanical effects. We consider a time-dependent administration of bacteria and analyze the impact of bacterial chemotaxis and killing rate. We show that active bacterial migration toward tumor hypoxic regions provides optimal infiltration and that high killing rates combined with high chemotactic values provide the smallest tumor volumes at the end of the treatment. We highlight the emergence of steady states in which a small population of bacteria is able to constrain tumor growth. Finally, we show that bacteria treatment works best in the case of tumors with high cellular proliferation and low oxygen consumption.

Spatially heterogeneous dynamics of cells in a growing tumor spheroid: comparison between theory and experiments

Collective cell movement, characterized by multiple cells that are in contact for substantial periods of time and undergo correlated motion, plays a central role in cancer and embryogenesis. Recent imaging experiments have provided time-dependent traces of individual cells, thus providing an unprecedented picture of tumor spheroid growth. By using simulations of a minimal cell model, we analyze the experimental data that map the movement of cells in a fibrosarcoma tumor spheroid embedded in a collagen matrix. Both simulations and experiments show that cells in the core of the spheroid exhibit subdiffusive glassy dynamics (mean square displacement, Δ(t) ≈ tα with α < 1), whereas cells in the periphery exhibit superdiffusive motion, Δ(t) ≈ tα with α > 1. The motion of most of the cells near the periphery is highly persistent and correlated directional motion due to cell doubling and apoptosis rates, thus explaining the observed superdiffusive behavior. The α values for cells in the core and periphery, extracted from simulations and experiments, are in near quantitative agreement with each other, which is surprising given that no parameter in the model was used to fit the measurements. The qualitatively different dynamics of cells in the core and periphery is captured by the fourth order susceptibility, introduced to characterize metastable states in glass forming systems. Analyses of the velocity autocorrelation of individual cells show remarkable spatial heterogeneity with no two cells exhibiting similar behavior. The prediction that α should depend on the location of the cells in the tumor is amenable to experimental testing. The highly heterogeneous dynamics of cells in the tumor spheroid provides a plausible mechanism for the origin of intratumor heterogeneity.

A quantitative high-resolution computational mechanics cell model for growing and regenerating tissues

Mathematical models are increasingly designed to guide experiments in biology, biotechnology, as well as to assist in medical decision making. They are in particular important to understand emergent collective cell behavior. For this purpose, the models, despite still abstractions of reality, need to be quantitative in all aspects relevant for the question of interest. This paper considers as showcase example the regeneration of liver after drug-induced depletion of hepatocytes, in which the surviving and dividing hepatocytes must squeeze in between the blood vessels of a network to refill the emerged lesions. Here, the cells' response to mechanical stress might significantly impact the regeneration process. We present a 3D high-resolution cell-based model integrating information from measurements in order to obtain a refined and quantitative understanding of the impact of cell-biomechanical effects on the closure of drug-induced lesions in liver. Our model represents each cell individually and is constructed by a discrete, physically scalable network of viscoelastic elements, capable of mimicking realistic cell deformation and supplying information at subcellular scales. The cells have the capability to migrate, grow, and divide, and the nature and parameters of their mechanical elements can be inferred from comparisons with optical stretcher experiments. Due to triangulation of the cell surface, interactions of cells with arbitrarily shaped (triangulated) structures such as blood vessels can be captured naturally. Comparing our simulations with those of so-called center-based models, in which cells have a largely rigid shape and forces are exerted between cell centers, we find that the migration forces a cell needs to exert on its environment to close a tissue lesion, is much smaller than predicted by center-based models. To stress generality of the approach, the liver simulations were complemented by monolayer and multicellular spheroid growth simulations. In summary, our model can give quantitative insight in many tissue organization processes, permits hypothesis testing in silico, and guide experiments in situations in which cell mechanics is considered important.

Three-dimensional tumor growth in time-varying chemical fields: a modeling framework and theoretical study

Background

Contemporary biological observations have revealed a large variety of mechanisms acting during the expansion of a tumor. However, there are still many qualitative and quantitative aspects of the phenomenon that remain largely unknown. In this context, mathematical and computational modeling appears as an invaluable tool providing the means for conducting in silico experiments, which are cheaper and less tedious than real laboratory experiments.

Results

This paper aims at developing an extensible and computationally efficient framework for in silico modeling of tumor growth in a 3-dimensional, inhomogeneous and time-varying chemical environment. The resulting model consists of a set of mathematically derived and algorithmically defined operators, each one addressing the effects of a particular biological mechanism on the state of the system. These operators may be extended or re-adjusted, in case a different set of starting assumptions or a different simulation scenario needs to be considered.

Conclusion

In silico modeling provides an alternative means for testing hypotheses and simulating scenarios for which exact biological knowledge remains elusive. However, finer tuning of pertinent methods presupposes qualitative and quantitative enrichment of available biological evidence. Validation in a strict sense would further require comprehensive, case-specific simulations and detailed comparisons with biomedical observations.

A checkpoint-oriented cell cycle simulation model

Modeling and in silico simulations are of major conceptual and applicative interest in studying the cell cycle and proliferation in eukaryotic cells. In this paper, we present a cell cycle checkpoint-oriented simulator that uses agent-based simulation modeling to reproduce the dynamics of a cancer cell population in exponential growth. Our in silico simulations were successfully validated by experimental in vitro supporting data obtained with HCT116 colon cancer cells. We demonstrated that this model can simulate cell confluence and the associated elongation of the G1 phase. Using nocodazole to synchronize cancer cells at mitosis, we confirmed the model predictivity and provided evidence of an additional and unexpected effect of nocodazole on the overall cell cycle progression. We anticipate that this cell cycle simulator will be a potential source of new insights and research perspectives.

Quantitative cell-based model predicts mechanical stress response of growing tumor spheroids over various growth conditions and cell lines

Model simulations indicate that the response of growing cell populations on mechanical stress follows the same functional relationship and is predictable over different cell lines and growth conditions despite experimental response curves look largely different. We develop a hybrid model strategy in which cells are represented by coarse-grained individual units calibrated with a high resolution cell model and parameterized by measurable biophysical and cell-biological parameters. Cell cycle progression in our model is controlled by volumetric strain, the latter being derived from a bio-mechanical relation between applied pressure and cell compressibility. After parameter calibration from experiments with mouse colon carcinoma cells growing against the resistance of an elastic alginate capsule, the model adequately predicts the growth curve in i) soft and rigid capsules, ii) in different experimental conditions where the mechanical stress is generated by osmosis via a high molecular weight dextran solution, and iii) for other cell types with different growth kinetics from the growth kinetics in absence of external stress. Our model simulation results suggest a generic, even quantitatively same, growth response of cell populations upon externally applied mechanical stress, as it can be quantitatively predicted using the same growth progression function.

Algorithm for a particle-based growth model for plant tissues

We have developed an algorithm for a particle-based model for the growth of plant tissues in three dimensions in which each cell is represented by a single particle, and connecting cell walls are represented as permanent bonds between particles. A sample of plant tissue is represented by a fixed network of bonded particles. If, and only if a cell divides, this network is updated locally. The update algorithm is implemented in a model where cell growth and division gives rise to forces between the cells, which are relaxed in steepest descent minimization. The same forces generate a pressure inside the cells, which moderates growth. The local nature of the algorithm makes it efficient computationally, so the model can deal with a large number of cells. We used the model to study the growth of plant tissues for a variety of model parameters, to show the viability of the algorithm.

3D multicellular model of shock wave-cell interaction

Understanding the interaction between shock waves and tissue is critical for advancing the use of shock waves for medical applications, such as cancer therapy. This work aims to study shock wave-cell interaction in a more realistic environment, relevant to in vitro and in vivo studies, by using 3D computational models of healthy and cancerous cells. The results indicate that for a single cell embedded in an extracellular environment, the cellular geometry does not influence significantly the membrane strain but does influence the von Mises stress. On the contrary, the presence of neighbouring cells has a strong effect on the cell response, by increasing fourfold both quantities. The membrane strain response of a cell converges with more than three neighbouring cell layers, indicating that a cluster of four layers of cells is sufficient to model the membrane strain in a large domain of tissue. However, a full 3D tissue model is needed if the stress evaluation is of main interest. A tumour mimicking multicellular spheroid model is also proposed to study mutual interaction between healthy and cancer cells and shows that cancer cells can be specifically targeted in an early stage tumour-mimicking environment.

STATEMENT OF SIGNIFICANCE

This work presents 3D computational models of shock-wave/cell interaction in a biophysically realistic environment using real cell morphology in tissue-mimicking phantoms and multicellular spheroids. Results show that cell morphology does not strongly influence the membrane strain but influences the von Mises stress. While the presence of neighbouring cells significantly increases the cell response, four cell layers are enough to capture the membrane strain change in tissue. However, a full tissue model is necessary if accurate stress analysis is needed. The work also shows that cancer cells can be specifically targeted in early stage tumour mimicking environment. This work is a step towards realistic modelling of shock-wave/cell interactions in tissues and provides insight on the use of 3D models for different scenarios.

Mathematical modeling of the proliferation gradient in multicellular tumor spheroids

MultiCellular Tumor Spheroids are 3D cell cultures that can accurately reproduce the behavior of solid tumors. It has been experimentally observed that large spheroids exhibit a decreasing gradient of proliferation from the periphery to the center of these multicellular 3D models: the proportion of proliferating cells is higher in the periphery while the non-proliferating quiescent cells increase in depth. In this paper, we propose to investigate the key mechanisms involved in the establishment of this gradient with a Partial Differential Equations model that mimics the experimental set-up of growing spheroids under different nutrients supply conditions. The model consists of mass balance equations on the two cell populations observed in the data: the proliferating cells and the quiescent cells. The spherical symmetry is used to rewrite the model in radial and relative coordinates. Thanks to a rigorous data postprocessing the model is then fit and compared quantitatively with the experimental quantification of the percentage of proliferating cells from EdU immunodetection on 2D spheroid cryosection images. The results of this calibration show that the proliferation gradient observed in spheroids can be quantitatively reproduced by our model.

The complex ecosystem in non small cell lung cancer invasion

Many tumors are characterized by genetic instability, producing an assortment of genetic variants of tumor cells called subclones. These tumors and their surrounding environments form complex multi-cellular ecosystems, where subclones compete for resources and cooperate to perform multiple tasks, including cancer invasion. Our recent empirical studies revealed existence of such distinct phenotypes of cancer cells, leaders and followers, in lung cancer. These two cellular subclones exchange a complex array of extracellular signals demonstrating a symbiotic relationship at the cellular level. Here, we develop a computational model of the microenvironment of the lung cancer ecosystem to explore how the interactions between subclones can advance or inhibit invasion. We found that, due to the complexity of the ecosystem, invasion may have very different dynamics characterized by the different levels of aggressiveness. By altering the signaling environment, we could alter the ecological relationship between the cell types and the overall ecosystem development. Competition between leader and follower cell populations (defined by the limited amount of resources), positive feedback within the leader cell population (controlled by the focal adhesion kinase and fibronectin signaling), and impact of the follower cells to the leaders (represented by yet undetermined proliferation signal) all had major effects on the outcome of the collective dynamics. Specifically, our analysis revealed a class of tumors (defined by the strengths of fibronectin signaling and competition) that are particularly sensitive to manipulations of the signaling environment. These tumors can undergo irreversible changes to the tumor ecosystem that outlast the manipulations of feedbacks and have a profound impact on invasive potential. Our study predicts a complex division of labor between cancer cell subclones and suggests new treatment strategies targeting signaling within the tumor ecosystem.

A multiscale model for heterogeneous tumor spheroid in vitro

In this paper, a novel multiscale method is proposed for the study of heterogeneous tumor spheroid growth in vitro. The entire tumor spheroid is described by an ellipsoid-based model while nutrient and other environmental factors are treated as continua. The ellipsoid-based discrete component is capable of incorporating mechanical effects and deformability, while keeping a minimum set of free variables to describe complex shape variations. Moreover, our purely cell-based description of tumor avoids the complex mutual conversion between a cell-based model and continuum model within a tumor, such as force and mass transformation. This advantage makes it highly suitable for the study of tumor spheroids in vitro whose size are normally less than 800 μm in diameter. In addition, our numerical scheme provides two computational options depending on tumor size. For a small or medium tumor spheroid, a three-dimensional (3D) numerical model can be directly applied. For a large spheroid, we suggest the use of a 3D-adapted 2D cross section configuration, which has not yet been explored in the literature, as an alternative for the theoretical investigation to bridge the gap between the 2D and 3D models. Our model and its implementations have been validated and applied to various studies given in the paper. The simulation results fit corresponding in vitro experimental observations very well.

Cell signaling heterogeneity is modulated by both cell-intrinsic and -extrinsic mechanisms: An integrated approach to understanding targeted therapy

During the last decade, our understanding of cancer cell signaling networks has significantly improved, leading to the development of various targeted therapies that have elicited profound but, unfortunately, short-lived responses. This is, in part, due to the fact that these targeted therapies ignore context and average out heterogeneity. Here, we present a mathematical framework that addresses the impact of signaling heterogeneity on targeted therapy outcomes. We employ a simplified oncogenic rat sarcoma (RAS)-driven mitogen-activated protein kinase (MAPK) and phosphoinositide 3-kinase-protein kinase B (PI3K-AKT) signaling pathway in lung cancer as an experimental model system and develop a network model of the pathway. We measure how inhibition of the pathway modulates protein phosphorylation as well as cell viability under different microenvironmental conditions. Training the model on this data using Monte Carlo simulation results in a suite of in silico cells whose relative protein activities and cell viability match experimental observation. The calibrated model predicts distributional responses to kinase inhibitors and suggests drug resistance mechanisms that can be exploited in drug combination strategies. The suggested combination strategies are validated using in vitro experimental data. The validated in silico cells are further interrogated through an unsupervised clustering analysis and then integrated into a mathematical model of tumor growth in a homogeneous and resource-limited microenvironment. We assess posttreatment heterogeneity and predict vast differences across treatments with similar efficacy, further emphasizing that heterogeneity should modulate treatment strategies. The signaling model is also integrated into a hybrid cellular automata (HCA) model of tumor growth in a spatially heterogeneous microenvironment. As a proof of concept, we simulate tumor responses to targeted therapies in a spatially segregated tissue structure containing tumor and stroma (derived from patient tissue) and predict complex cell signaling responses that suggest a novel combination treatment strategy.

A signaling pathway is a network of molecules in a cell that is typically initiated by stimuli (e.g., microenvironmental cues) acting on receptors and internal signaling molecules to determine cell fate. Signaling pathways in cancer cells are different from those in normal cells, and this difference helps cancer cells to grow and thrive indefinitely. Drugs that target the aberrant signaling pathways in cancer cells (often referred to as targeted therapy) are promising for improving treatment outcomes of many different cancers in patients. However, most patients eventually develop resistance to these drugs. Resistance may already be present in the tumor or may emerge via mutation or via microenvironmental mediation. Tumor heterogeneity, which is characterized by subtle or dramatic differences among tumor cells, plays a key role in the development of drug resistance. Some tumor cells respond well to therapy, while others may adapt to the stress induced by the drug within the microenvironment. Moreover, removal of drug-sensitive cells may result in the competitive release of drug-resistant cells. Here, we present mathematical models to assess the impact of heterogeneity in signaling pathways within tumor cells on the outcomes of targeted therapy. We consider a simplified version of two well-known signaling pathways that modulate the growth of lung cancer cells. By using different targeted therapies, we quantify the effect of pathway inhibition on protein activity and cell viability and developed a mathematical model of the network, which is trained to reproduce these data and to develop a panel of heterogeneous in silico cells. The model predicts potential mechanisms of drug resistance and proposes combination therapies that are effective across the panel. We validate these combination therapies experimentally using the lung cancer cells and integrated the in silico cells into a computational lung tissue model that explicitly captures the microenvironment of lung cancer. Our results suggest that heterogeneity in both the tumor and microenvironment impacts treatment response in different ways and suggest a novel combination therapy for a better response.

A Brownian dynamics tumor progression simulator with application to glioblastoma

Tumor progression modeling offers the potential to predict tumor-spreading behavior to improve prognostic accuracy and guide therapy development. Common simulation methods include continuous reaction-diffusion (RD) approaches that capture mean spatio-temporal tumor spreading behavior and discrete agent-based (AB) approaches which capture individual cell events such as proliferation or migration. The brain cancer glioblastoma (GBM) is especially appropriate for such proliferation-migration modeling approaches because tumor cells seldom metastasize outside of the central nervous system and cells are both highly proliferative and migratory. In glioblastoma research, current RD estimates of proliferation and migration parameters are derived from computed tomography or magnetic resonance images. However, these estimates of glioblastoma cell migration rates, modeled as a diffusion coefficient, are approximately 1-2 orders of magnitude larger than single-cell measurements in animal models of this disease. To identify possible sources for this discrepancy, we evaluated the fundamental RD simulation assumptions that cells are point-like structures that can overlap. To give cells physical size (~10 μm), we used a Brownian dynamics approach that simulates individual single-cell diffusive migration, growth, and proliferation activity via a gridless, off-lattice, AB method where cells can be prohibited from overlapping each other. We found that for realistic single-cell parameter growth and migration rates, a non-overlapping model gives rise to a jammed configuration in the center of the tumor and a biased outward diffusion of cells in the tumor periphery, creating a quasi-ballistic advancing tumor front. The simulations demonstrate that a fast-progressing tumor can result from minimally diffusive cells, but at a rate that is still dependent on single-cell diffusive migration rates. Thus, modeling with the assumption of physically-grounded volume conservation can account for the apparent discrepancy between estimated and measured diffusion of GBM cells and provide a new theoretical framework that naturally links single-cell growth and migration dynamics to tumor-level progression.

Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-[Formula: see text]B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53-Mdm2, NF-[Formula: see text]B) and through the use of high-performance computing be capable of simulating up to [Formula: see text] cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.

Multiscale image analysis reveals structural heterogeneity of the cell microenvironment in homotypic spheroids

Three-dimensional multicellular aggregates such as spheroids provide reliable in vitro substitutes for tissues. Quantitative characterization of spheroids at the cellular level is fundamental. We present the first pipeline that provides three-dimensional, high-quality images of intact spheroids at cellular resolution and a comprehensive image analysis that completes traditional image segmentation by algorithms from other fields. The pipeline combines light sheet-based fluorescence microscopy of optically cleared spheroids with automated nuclei segmentation (F score: 0.88) and concepts from graph analysis and computational topology. Incorporating cell graphs and alpha shapes provided more than 30 features of individual nuclei, the cellular neighborhood and the spheroid morphology. The application of our pipeline to a set of breast carcinoma spheroids revealed two concentric layers of different cell density for more than 30,000 cells. The thickness of the outer cell layer depends on a spheroid’s size and varies between 50% and 75% of its radius. In differently-sized spheroids, we detected patches of different cell densities ranging from 5 × 10⁵ to 1 × 10⁶cells/mm³. Since cell density affects cell behavior in tissues, structural heterogeneities need to be incorporated into existing models. Our image analysis pipeline provides a multiscale approach to obtain the relevant data for a system-level understanding of tissue architecture.

Parallelization and High-Performance Computing Enables Automated Statistical Inference of Multi-scale Models

Mechanistic understanding of multi-scale biological processes, such as cell proliferation in a changing biological tissue, is readily facilitated by computational models. While tools exist to construct and simulate multi-scale models, the statistical inference of the unknown model parameters remains an open problem. Here, we present and benchmark a parallel approximate Bayesian computation sequential Monte Carlo (pABC SMC) algorithm, tailored for high-performance computing clusters. pABC SMC is fully automated and returns reliable parameter estimates and confidence intervals. By running the pABC SMC algorithm for ∼10⁶ hr, we parameterize multi-scale models that accurately describe quantitative growth curves and histological data obtained in vivo from individual tumor spheroid growth in media droplets. The models capture the hybrid deterministic-stochastic behaviors of 10⁵-10⁶ of cells growing in a 3D dynamically changing nutrient environment. The pABC SMC algorithm reliably converges to a consistent set of parameters. Our study demonstrates a proof of principle for robust, data-driven modeling of multi-scale biological systems and the feasibility of multi-scale model parameterization through statistical inference.

3D hybrid modelling of vascular network formation

We develop an off-lattice, agent-based model to describe vasculogenesis, the de novo formation of blood vessels from endothelial progenitor cells during development. The endothelial cells that comprise our vessel network are viewed as linearly elastic spheres that move in response to the forces they experience. We distinguish two types of endothelial cells: vessel elements are contained within the network and tip cells are located at the ends of vessels. Tip cells move in response to mechanical forces caused by interactions with neighbouring vessel elements and the local tissue environment, chemotactic forces and a persistence force which accounts for their tendency to continue moving in the same direction. Vessel elements are subject to similar mechanical forces but are insensitive to chemotaxis. An angular persistence force representing interactions with the local tissue is introduced to stabilise buckling instabilities caused by cell proliferation. Only vessel elements proliferate, at rates which depend on their degree of stretch: elongated elements have increased rates of proliferation, and compressed elements have reduced rates. Following division, the fate of the new cell depends on the local mechanical environment: the probability of forming a new sprout is increased if the parent vessel is highly compressed and the probability of being incorporated into the parent vessel increased if the parent is stretched. Simulation results reveal that our hybrid model can reproduce the key qualitative features of vasculogenesis. Extensive parameter sensitivity analyses show that significant changes in network size and morphology are induced by varying the chemotactic sensitivity of tip cells, and the sensitivities of the proliferation rate and the sprouting probability to mechanical stretch. Varying the chemotactic sensitivity directly influences the directionality of the networks. The degree of branching, and thereby the density of the networks, is influenced by the sprouting probability. Glyphs that simultaneously depict several network properties are introduced to show how these and other network quantities change over time and also as model parameters vary. We also show how equivalent glyphs constructed from in vivo data could be used to discriminate between normal and tumour vasculature and, in the longer term, for model validation. We conclude that our biomechanical hybrid model can generate vascular networks that are qualitatively similar to those generated from in vitro and in vivo experiments.

3D tumor spheroids: an overview on the tools and techniques used for their analysis

In comparison with 2D cell culture models, 3D spheroids are able to accurately mimic some features of solid tumors, such as their spatial architecture, physiological responses, secretion of soluble mediators, gene expression patterns and drug resistance mechanisms. These unique characteristics highlight the potential of 3D cellular aggregates to be used as in vitro models for screening new anticancer therapeutics, both at a small and large scale. Nevertheless, few reports have focused on describing the tools and techniques currently available to extract significant biological data from these models. Such information will be fundamental to drug and therapeutic discovery process using 3D cell culture models. The present review provides an overview of the techniques that can be employed to characterize and evaluate the efficacy of anticancer therapeutics in 3D tumor spheroids.

Inoculation density and nutrient level determine the formation of mushroom-shaped structures in Pseudomonas aeruginosa biofilms

Pseudomonas aeruginosa often colonises immunocompromised patients and the lungs of cystic fibrosis patients. It exhibits resistance to many antibiotics by forming biofilms, which makes it hard to eliminate. P. aeruginosa biofilms form mushroom-shaped structures under certain circumstances. Bacterial motility and the environment affect the eventual mushroom morphology. This study provides an agent-based model for the bacterial dynamics and interactions influencing bacterial biofilm shape. Cell motility in the model relies on recently published experimental data. Our simulations show colony formation by immotile cells. Motile cells escape from a single colony by nutrient chemotaxis and hence no mushroom shape develops. A high number density of non-motile colonies leads to migration of motile cells onto the top of the colonies and formation of mushroom-shaped structures. This model proposes that the formation of mushroom-shaped structures can be predicted by parameters at the time of bacteria inoculation. Depending on nutrient levels and the initial number density of stalks, mushroom-shaped structures only form in a restricted regime. This opens the possibility of early manipulation of spatial pattern formation in bacterial colonies, using environmental factors.

Cancer therapeutic potential of combinatorial immuno- and vasomodulatory interventions

Currently, most of the basic mechanisms governing tumour-immune system interactions, in combination with modulations of tumour-associated vasculature, are far from being completely understood. Here, we propose a mathematical model of vascularized tumour growth, where the main novelty is the modelling of the interplay between functional tumour vasculature and effector cell recruitment dynamics. Parameters are calibrated on the basis of different in vivo immunocompromised Rag1(-/-) and wild-type (WT) BALB/c murine tumour growth experiments. The model analysis supports that tumour vasculature normalization can be a plausible and effective strategy to treat cancer when combined with appropriate immunostimulations. We find that improved levels of functional tumour vasculature, potentially mediated by normalization or stress alleviation strategies, can provide beneficial outcomes in terms of tumour burden reduction and growth control. Normalization of tumour blood vessels opens a therapeutic window of opportunity to augment the antitumour immune responses, as well as to reduce intratumoral immunosuppression and induced hypoxia due to vascular abnormalities. The potential success of normalizing tumour-associated vasculature closely depends on the effector cell recruitment dynamics and tumour sizes. Furthermore, an arbitrary increase in the initial effector cell concentration does not necessarily imply better tumour control. We evidence the existence of an optimal concentration range of effector cells for tumour shrinkage. Based on these findings, we suggest a theory-driven therapeutic proposal that optimally combines immuno- and vasomodulatory interventions.

A Mechanistic Collective Cell Model for Epithelial Colony Growth and Contact Inhibition

We present a mechanistic hybrid continuum-discrete model to simulate the dynamics of epithelial cell colonies. Collective cell dynamics are modeled using continuum equations that capture plastic, viscoelastic, and elastic deformations in the clusters while providing single-cell resolution. The continuum equations can be viewed as a coarse-grained version of previously developed discrete models that treat epithelial clusters as a two-dimensional network of vertices or stochastic interacting particles and follow the framework of dynamic density functional theory appropriately modified to account for cell size and shape variability. The discrete component of the model implements cell division and thus influences cell size and shape that couple to the continuum component. The model is validated against recent in vitro studies of epithelial cell colonies using Madin-Darby canine kidney cells. In good agreement with experiments, we find that mechanical interactions and constraints on the local expansion of cell size cause inhibition of cell motion and reductive cell division. This leads to successively smaller cells and a transition from exponential to quadratic growth of the colony that is associated with a constant-thickness rim of growing cells at the cluster edge, as well as the emergence of short-range ordering and solid-like behavior. A detailed analysis of the model reveals a scale invariance of the growth and provides insight into the generation of stresses and their influence on the dynamics of the colonies. Compared to previous models, our approach has several advantages: it is independent of dimension, it can be parameterized using classical elastic properties (Poisson's ratio and Young's modulus), and it can easily be extended to incorporate multiple cell types and general substrate geometries.

Inferring Growth Control Mechanisms in Growing Multi-cellular Spheroids of NSCLC Cells from Spatial-Temporal Image Data

We develop a quantitative single cell-based mathematical model for multi-cellular tumor spheroids (MCTS) of SK-MES-1 cells, a non-small cell lung cancer (NSCLC) cell line, growing under various nutrient conditions: we confront the simulations performed with this model with data on the growth kinetics and spatial labeling patterns for cell proliferation, extracellular matrix (ECM), cell distribution and cell death. We start with a simple model capturing part of the experimental observations. We then show, by performing a sensitivity analysis at each development stage of the model that its complexity needs to be stepwise increased to account for further experimental growth conditions. We thus ultimately arrive at a model that mimics the MCTS growth under multiple conditions to a great extent. Interestingly, the final model, is a minimal model capable of explaining all data simultaneously in the sense, that the number of mechanisms it contains is sufficient to explain the data and missing out any of its mechanisms did not permit fit between all data and the model within physiological parameter ranges. Nevertheless, compared to earlier models it is quite complex i.e., it includes a wide range of mechanisms discussed in biological literature. In this model, the cells lacking oxygen switch from aerobe to anaerobe glycolysis and produce lactate. Too high concentrations of lactate or too low concentrations of ATP promote cell death. Only if the extracellular matrix density overcomes a certain threshold, cells are able to enter the cell cycle. Dying cells produce a diffusive growth inhibitor. Missing out the spatial information would not permit to infer the mechanisms at work. Our findings suggest that this iterative data integration together with intermediate model sensitivity analysis at each model development stage, provide a promising strategy to infer predictive yet minimal (in the above sense) quantitative models of tumor growth, as prospectively of other tissue organization processes. Importantly, calibrating the model with two nutriment-rich growth conditions, the outcome for two nutriment-poor growth conditions could be predicted. As the final model is however quite complex, incorporating many mechanisms, space, time, and stochastic processes, parameter identification is a challenge. This calls for more efficient strategies of imaging and image analysis, as well as of parameter identification in stochastic agent-based simulations.

We here present how to parameterize a mathematical agent-based model of growing MCTS almost completely from experimental data. MCTS show a similar establishment of pathophysiological gradients and concentric arrangement of heterogeneous cell populations as found in avascular tumor nodules. We build a process chain of imaging, image processing and analysis, and mathematical modeling. In this model, each individual cell is represented by an agent populating one site of a three dimensional un-structured lattice. The spatio-temporal multi-cellular behavior, including migration, growth, division, death of each cell, is considered by a stochastic process, simulated numerically by the Gillespie algorithm. Processes on the molecular scale are described by deterministic partial differential equations for molecular concentrations, coupled to intracellular and cellular decision processes. The parameters of the multi-scale model are inferred from comparisons to the growth kinetics and from image analysis of spheroid cryosections stained for cell death, proliferation and collagen IV. Our final model assumes ATP to be the critical resource that cells try to keep constant over a wide range of oxygen and glucose medium concentrations, by switching between aerobic and anaerobic metabolism. Besides ATP, lactate is shown to be a possible explanation for the control of the necrotic core size. Direct confrontation of the model simulation results with image data on the spatial profiles of cell proliferation, ECM distribution and cell death, indicates that in addition, the effects of ECM and waste factors have to be added to explain the data. Hence the model is a tool to identify likely mechanisms at work that may subsequently be studied experimentally, proposing a model-guided experimental strategy.

A cell-centered, agent-based framework that enables flexible environment granularities

BACKGROUND

Mechanistic explanations of cell-level phenomena typically adopt an observer perspective. Explanations developed from a cell's perspective may offer new insights. Agent-based models lend themselves to model from an individual perspective, and existing agent-based models generally utilize a regular lattice-based environment. A framework which utilizes a cell's perspective in an off-lattice environment could improve the overall understanding of biological phenomena.

RESULTS

We present an agent-based, discrete event framework, with a demonstrative focus on biomimetic agents. The framework was first developed in 2-dimensions and then extended, with a subset of behaviors, to 3-dimensions. The framework is expected to facilitate studies of more complex biological phenomena through exploitation of a dynamic Delaunay and Voronoi off-lattice environment. We used the framework to model biological cells and to specifically demonstrate basic biological cell behaviors in two- and three-dimensional space. Potential use cases are highlighted, suggesting the utility of the framework in various scenarios.

CONCLUSIONS

The framework presented in this manuscript expands on existing cell- and agent-centered methods by offering a new perspective in an off-lattice environment. As the demand for biomimetic models grows, the demand for new methods, such as the presented Delaunay and Voronoi framework, is expected to increase.

Lateral assembly of N-cadherin drives tissue integrity by stabilizing adherens junctions

Cadherin interactions ensure the correct registry and anchorage of cells during tissue formation. Along the plasma membrane, cadherins form inter-junctional lattices via cis- and trans-dimerization. While structural studies have provided models for cadherin interactions, the molecular nature of cadherin binding in vivo remains unexplored. We undertook a multi-disciplinary approach combining live cell imaging of three-dimensional cell assemblies (spheroids) with a computational model to study the dynamics of N-cadherin interactions. Using a loss-of-function strategy, we demonstrate that each N-cadherin interface plays a distinct role in spheroid formation. We found that cis-dimerization is not a prerequisite for trans-interactions, but rather modulates trans-interfaces to ensure tissue stability. Using a model of N-cadherin junction dynamics, we show that the absence of cis-interactions results in low junction stability and loss of tissue integrity. By quantifying the binding and unbinding dynamics of the N-cadherin binding interfaces, we determined that mutating either interface results in a 10-fold increase in the dissociation constant. These findings provide new quantitative information on the steps driving cadherin intercellular adhesion and demonstrate the role of cis-interactions in junction stability.

A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity

Most cancers in humans are large, measuring centimetres in diameter, and composed of many billions of cells. An equivalent mass of normal cells would be highly heterogeneous as a result of the mutations that occur during each cell division. What is remarkable about cancers is that virtually every neoplastic cell within a large tumour often contains the same core set of genetic alterations, with heterogeneity confined to mutations that emerge late during tumour growth. How such alterations expand within the spatially constrained three-dimensional architecture of a tumour, and come to dominate a large, pre-existing lesion, has been unclear. Here we describe a model for tumour evolution that shows how short-range dispersal and cell turnover can account for rapid cell mixing inside the tumour. We show that even a small selective advantage of a single cell within a large tumour allows the descendants of that cell to replace the precursor mass in a clinically relevant time frame. We also demonstrate that the same mechanisms can be responsible for the rapid onset of resistance to chemotherapy. Our model not only provides insights into spatial and temporal aspects of tumour growth, but also suggests that targeting short-range cellular migratory activity could have marked effects on tumour growth rates.

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.

Dynamic mechanical finite element model of biological cells for studying cellular pattern formation

Understanding the geometric, topologic, and mechanical properties of cells and their interactions is critical for studying tissue pattern formation and organ development. Computational model and tools for simulating cell pattern formation have broad implications in studying embryogenesis, blood-vessel development, tissue regeneration, and tumor growth. Although a number of cell modeling methods exist, they do not simultaneously account for detailed cellular shapes as well as dynamic changes in cell geometry and topology. Here we describe a dynamic finite element cell model (dFEMC) for studying populations of cells and tissue development. By incorporating details of cell shape, cell growth and shrinkage, cell birth and death, cell division and fusion, our method can model realistically a variety problems of cell pattern formation. We give two examples of applying our method to the study of cell fusion and cell apoptosis. The dFEMC model developed here provides a general computational framework for studying dynamics pattern formation of tissue.

A computational model of nuclear self-organisation in syncytial embryos

Syncytial embryos develop through cycles of nuclear division and rearrangement within a common cytoplasm. A paradigm example is Drosophila melanogaster in which nuclei form an ordered array in the embryo surface over cell cycles 10-13. This ordering process is assumed to be essential for subsequent cellularisation. Using quantitative tissue analysis, it has previously been shown that the regrowth of actin and microtubule networks after nuclear division generates reordering forces that counteract its disordering effect (Kanesaki et al., 2011). We present here an individual-based computer simulation modelling the nuclear dynamics. In contrast to similar modelling approaches e.g. epithelial monolayers or tumour spheroids, we focus not on the spatial dependence, but rather on the time-dependence of the interaction laws. We show that appropriate phenomenological inter-nuclear force laws reproduce the experimentally observed dynamics provided that the cytoskeletal network regrows sufficiently quickly after mitosis. Then repulsive forces provided by the actin system are necessary and sufficient to regain the observed level of order in the system, after the strong disruption resulting from cytoskeletal network disassembly and spindle formation. We also observe little mixing of nuclei through cell cycles. Our study highlights the importance of the dynamics of cytoskeletal forces during this critical phase of syncytial development and emphasises the need for real-time experimental data at high temporal resolution.