Coherent modelling switch between pointwise and distributed representations of cell aggregates

A Multicellular Mechanochemical Model to Investigate Tumor Microenvironment Remodeling and Pre-Metastatic Niche Formation

Introduction

Colorectal cancer (CRC) is a major cause of cancer related deaths in the United States, with CRC metastasis to the liver being a common occurrence. The development of an optimal metastatic environment is essential process prior to tumor metastasis. This process, called pre-metastatic niche (PMN) formation, involves activation of key resident liver cells, including fibroblast-like stellate cells and macrophages such as Kupffer cells. Tumor-mediated factors introduced to this environment transform resident cells that secrete additional growth factors and remodel the extracellular matrix (ECM), which is thought to promote tumor colonization and metastasis in the secondary environment.

Methods

To investigate the underlying mechanisms of these dynamics, we developed a multicellular computational model to characterize the spatiotemporal dynamics of the PMN formation in tissue. This modeling framework integrates intracellular and extracellular signaling, and traction and junctional forces into a Cellular Potts model, and represents multiple cell types with varying levels of cellular activation. We perform numerical experiments to investigate the role of key factors in PMN formation and tumor invasiveness, including growth factor concentration, timing of tumor arrival, relative composition of resident cells, and the size of invading tumor cluster.

Results

These parameter studies identified growth factor availability and ECM concentration in the environment as two of the key determinants of tumor invasiveness. We further predict that both the ECM concentration potential and growth factor sensitivity of the stellate cells are key drivers of the PMN formation and associated ECM concentration.

Conclusions

Overall, this modeling framework represents a significant step towards simulating cancer metastasis and investigating the role of key factors on PMN formation.

Supplementary Information

The online version contains supplementary material available at 10.1007/s12195-024-00831-0.

A Genuinely Hybrid, Multiscale 3D Cancer Invasion and Metastasis Modelling Framework

We introduce in this paper substantial enhancements to a previously proposed hybrid multiscale cancer invasion modelling framework to better reflect the biological reality and dynamics of cancer. These model updates contribute to a more accurate representation of cancer dynamics, they provide deeper insights and enhance our predictive capabilities. Key updates include the integration of porous medium-like diffusion for the evolution of Epithelial-like Cancer Cells and other essential cellular constituents of the system, more realistic modelling of Epithelial-Mesenchymal Transition and Mesenchymal-Epithelial Transition models with the inclusion of Transforming Growth Factor beta within the tumour microenvironment, and the introduction of Compound Poisson Process in the Stochastic Differential Equations that describe the migration behaviour of the Mesenchymal-like Cancer Cells. Another innovative feature of the model is its extension into a multi-organ metastatic framework. This framework connects various organs through a circulatory network, enabling the study of how cancer cells spread to secondary sites.

Multi-Cue Kinetic Model with Non-Local Sensing for Cell Migration on a Fiber Network with Chemotaxis

Cells perform directed motion in response to external stimuli that they detect by sensing the environment with their membrane protrusions. Precisely, several biochemical and biophysical cues give rise to tactic migration in the direction of their specific targets. Thus, this defines a multi-cue environment in which cells have to sort and combine different, and potentially competitive, stimuli. We propose a non-local kinetic model for cell migration in which cell polarization is influenced simultaneously by two external factors: contact guidance and chemotaxis. We propose two different sensing strategies, and we analyze the two resulting transport kinetic models by recovering the appropriate macroscopic limit in different regimes, in order to observe how the cell size, with respect to the variation of both external fields, influences the overall behavior. This analysis shows the importance of dealing with hyperbolic models, rather than drift-diffusion ones. Moreover, we numerically integrate the kinetic transport equations in a two-dimensional setting in order to investigate qualitatively various scenarios. Finally, we show how our setting is able to reproduce some experimental results concerning the influence of topographical and chemical cues in directing cell motility.

An agent-based approach for modelling collective dynamics in animal groups distinguishing individual speed and orientation

Collective dynamics in animal groups is a challenging theme for the modelling community, being treated with a wide range of approaches. This topic is here tackled by a discrete model. Entering in more details, each agent, represented by a material point, is assumed to move following a first-order Newtonian law, which distinguishes speed and orientation. In particular, the latter results from the balance of a given set of behavioural stimuli, each of them defined by a direction and a weight, that quantifies its relative importance. A constraint on the sum of the weights then avoids implausible simultaneous maximization/minimization of all movement traits. Our framework is based on a minimal set of rules and parameters and is able to capture and classify a number of collective group dynamics emerging from different individual preferred behaviour, which possibly includes attractive, repulsive and alignment stimuli. In the case of a system of animals subjected only to the first two behavioural inputs, we also show how analytical arguments allow us to a priori relate the equilibrium interparticle spacing to critical model coefficients. Our approach is then extended to account for the presence of predators with different hunting strategies, which impact on the behaviour of a prey population. Hints for model refinement and applications are finally given in the conclusive part of the article. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Modelling chase-and-run migration in heterogeneous populations

Cell migration is crucial for many physiological and pathological processes. During embryogenesis, neural crest cells undergo coordinated epithelial to mesenchymal transformations and migrate towards various forming organs. Here we develop a computational model to understand how mutual interactions between migrating neural crest cells (NCs) and the surrounding population of placode cells (PCs) generate coordinated migration. According to experimental findings, we implement a minimal set of hypotheses, based on a coupling between chemotactic movement of NCs in response to a placode-secreted chemoattractant (Sdf1) and repulsion induced from contact inhibition of locomotion (CIL), triggered by heterotypic NC–PC contacts. This basic set of assumptions is able to semi-quantitatively recapitulate experimental observations of the characteristic multispecies phenomenon of “chase-and-run”, where the colony of NCs chases an evasive PC aggregate. The model further reproduces a number of in vitro manipulations, including full or partial disruption of NC chemotactic migration and selected mechanisms coordinating the CIL phenomenon. Finally, we provide various predictions based on altering other key components of the model mechanisms.

Kinetic models with non-local sensing determining cell polarization and speed according to independent cues

Cells move by run and tumble, a kind of dynamics in which the cell alternates runs over straight lines and re-orientations. This erratic motion may be influenced by external factors, like chemicals, nutrients, the extra-cellular matrix, in the sense that the cell measures the external field and elaborates the signal eventually adapting its dynamics. We propose a kinetic transport equation implementing a velocity-jump process in which the transition probability takes into account a double bias, which acts, respectively, on the choice of the direction of motion and of the speed. The double bias depends on two different non-local sensing cues coming from the external environment. We analyze how the size of the cell and the way of sensing the environment with respect to the variation of the external fields affect the cell population dynamics by recovering an appropriate macroscopic limit and directly integrating the kinetic transport equation. A comparison between the solutions of the transport equation and of the proper macroscopic limit is also performed.

A particle model analysing the behavioural rules underlying the collective flight of a bee swarm towards the new nest

The swarming of a bee colony is guided by a small group of scout individuals, which are informed of the target destination (the new nest). However, little is known on the underlying mechanisms, i.e. on how the information is passed within the population. In this respect, we here present a discrete mathematical model to investigate these aspects. In particular, each bee, represented by a material point, is assigned its status within the colony and set to move according to individual strategies and social interactions. More specifically, we propose alternative assumptions on the flight synchronization mechanism of uninformed individuals and on the characteristic dynamics of the scout insects. Numerical realizations then point out the combinations of behavioural hypotheses resulting in collective productive movement. An analysis of the role of the scout bee percentage and of the phenomenology of the swarm in domains with structural elements is finally performed.

Non-local Parabolic and Hyperbolic Models for Cell Polarisation in Heterogeneous Cancer Cell Populations

Tumours consist of heterogeneous populations of cells. The sub-populations can have different features, including cell motility, proliferation and metastatic potential. The interactions between clonal sub-populations are complex, from stable coexistence to dominant behaviours. The cell–cell interactions, i.e. attraction, repulsion and alignment, processes critical in cancer invasion and metastasis, can be influenced by the mutation of cancer cells. In this study, we develop a mathematical model describing cancer cell invasion and movement for two polarised cancer cell populations with different levels of mutation. We consider a system of non-local hyperbolic equations that incorporate cell–cell interactions in the speed and the turning behaviour of cancer cells, and take a formal parabolic limit to transform this model into a non-local parabolic model. We then investigate the possibility of aggregations to form, and perform numerical simulations for both hyperbolic and parabolic models, comparing the patterns obtained for these models.

A discrete particle model reproducing collective dynamics of a bee swarm

In this article, we present a microscopic discrete mathematical model describing collective dynamics of a bee swarm. More specifically, each bee is set to move according to individual strategies and social interactions, the former involving the desire to reach a target destination, the latter accounting for repulsive/attractive stimuli and for alignment processes. The insects tend in fact to remain sufficiently close to the rest of the population, while avoiding collisions, and they are able to track and synchronize their movement to the flight of a given set of neighbors within their visual field. The resulting collective behavior of the bee cloud therefore emerges from non-local short/long-range interactions. Differently from similar approaches present in the literature, we here test different alignment mechanisms (i.e., based either on an Euclidean or on a topological neighborhood metric), which have an impact also on the other social components characterizing insect behavior. A series of numerical realizations then shows the phenomenology of the swarm (in terms of pattern configuration, collective productive movement, and flight synchronization) in different regions of the space of free model parameters (i.e., strength of attractive/repulsive forces, extension of the interaction regions). In this respect, constraints in the possible variations of such coefficients are here given both by reasonable empirical observations and by analytical results on some stability characteristics of the defined pairwise interaction kernels, which have to assure a realistic crystalline configuration of the swarm. An analysis of the effect of unconscious random fluctuations of bee dynamics is also provided.