Stress generation, relaxation and size control in confined tumor growth

Tissue stresses caused by invasive tumour: a biomechanical model

Malignant tumorigenesis is a complex process involving growth, invasion and mechanical deformation of a cancerous tissue. In this paper, a biomechanical model is proposed to couple the mechanical and biological mechanisms governing invasive tumour development. As an example, this model is applied to investigate the spatio-temporal evolution of tissue stresses in an invasive tumour spheroid and its host tissue. I show that cancer invasiveness lowers the compressive tissue stresses and blurs the stress distribution across the cancerous-normal tissue boundary, both consistent with experimental observations. Importantly, with the steady propagation of the cancerous region driven by persistent cancer invasion, tumour stresses are predicted to saturate rather than keep increasing as in benign tumour growth. The model is further used to analyse the deformation and stress state of a cancerous tissue being cut into two pieces, and reproduces the bulge of the cut surface observed in experiments. I hope this study can pave the way for the quantitative evaluation of mechanical states in cancer.

Unjamming Transition as a Paradigm for Biomechanical Control of Cancer Metastasis

Tumor metastasis is a complex phenomenon that poses significant challenges to current cancer therapeutics. While the biochemical signaling involved in promoting motile phenotypes is well understood, the role of biomechanical interactions has recently begun to be incorporated into models of tumor cell migration. Specifically, we propose the unjamming transition, adapted from physical paradigms describing the behavior of granular materials, to better discern the transition toward an invasive phenotype. In this review, we introduce the jamming transition broadly and narrow our discussion to the different modes of 3D tumor cell migration that arise. Then we discuss the mechanical interactions between tumor cells and their neighbors, along with the interactions between tumor cells and the surrounding extracellular matrix. We center our discussion on the interactions that induce a motile state or unjamming transition in these contexts. By considering the interplay between biochemical and biomechanical signaling in tumor cell migration, we can advance our understanding of biomechanical control in cancer metastasis.

A morpho-viscoelasticity theory for growth in proliferating aggregates

Despite significant research efforts in the continuum modeling of biological growth, certain aspects have been overlooked. For instance, numerous investigations have examined the influence of morphogenetic cell behaviors, like division and intercalation, on the mechanical response of passive (non-growing) tissues. Yet, their impact on active growth dynamics remains inadequately explored. A key reason for this inadequacy stems from challenges in the continuum treatment of cell-level processes. While some coarse-grained models have been proposed to address these shortcomings, a focus on cell division and cell expansion has been missing, rendering them unusable when it comes to modeling growth. Moreover, existing studies are limited to two-dimensional tissues and are yet to be formally extended to three-dimensional multicellular systems. To address these limitations, we here present a generalized multiscale model for three-dimensional aggregates that accounts for complex morphogenetic movements that include division, expansion, and intercalation. The proposed continuum theory thus allows for a comprehensive exploration into the growth and dissipation mechanics of proliferating aggregates, such as spheroids and organoids.

Mechanics and Morphology of Proliferating Cell Collectives with Self-Inhibiting Growth

We study the dynamics of proliferating cell collectives whose microscopic constituents' growth is inhibited by macroscopic growth-induced stress. Discrete particle simulations of a growing collective show the emergence of concentric-ring patterns in cell size whose spatiotemporal structure is closely tied to the individual cell's stress response. Motivated by these observations, we derive a multiscale continuum theory whose parameters map directly to the discrete model. Analytical solutions of this theory show the concentric patterns arise from anisotropically accumulated resistance to growth over many cell cycles. This Letter shows how purely mechanical processes can affect the internal patterning and morphology of cell collectives, and provides a concise theoretical framework for connecting the micro- to macroscopic dynamics of proliferating matter.

Matrix confinement modulates 3D spheroid sorting and burst-like collective migration

While it is known that cells with differential adhesion tend to segregate and preferentially sort, the physical forces governing sorting and invasion in heterogeneous tumors remain poorly understood. To investigate this, we tune matrix confinement, mimicking changes in the stiffness and confinement of the tumor microenvironment, to explore how physical confinement influences individual and collective cell migration in 3D spheroids. High levels of confinement lead to cell sorting while reducing matrix confinement triggers the collective fluidization of cell motion. Cell sorting, which depends on cell-cell adhesion, is crucial to this phenomenon. Burst-like migration does not occur for spheroids that have not undergone sorting, regardless of the degree of matrix confinement. Using computational Self-Propelled Voronoi modeling, we show that spheroid sorting and invasion into the matrix depend on the balance between cell-generated forces and matrix resistance. The findings support a model where matrix confinement modulates 3D spheroid sorting and unjamming in an adhesion-dependent manner, providing insights into the mechanisms of cell sorting and migration in the primary tumor and toward distant metastatic sites. STATEMENT OF SIGNIFICANCE: The mechanical properties of the tumor microenvironment significantly influence cancer cell migration within the primary tumor, yet how these properties affect intercellular interactions in heterogeneous tumors is not well understood. By utilizing calcium and calcium chelators, we dynamically alter collagen-alginate hydrogel stiffness and investigate tumor cell behavior within co-culture spheroids in response to varying degrees of matrix confinement. High confinement is found to trigger cell sorting while reducing confinement for sorted spheroids facilitates collective cell invasion. Notably, without prior sorting, spheroids do not exhibit burst-like migration, regardless of confinement levels. This work establishes that matrix confinement and intercellular adhesion regulate 3D spheroid dynamics, offering insights into cellular organization and migration within the primary tumor.

Matrix confinement modulates 3D spheroid sorting and burst-like collective migration

While it is known that cells with differential adhesion tend to segregate and preferentially sort, the physical forces governing sorting and invasion in heterogeneous tumors remain poorly understood. To investigate this, we tune matrix confinement, mimicking changes in the stiffness and confinement of the tumor microenvironment, to explore how physical confinement influences individual and collective cell migration in 3D spheroids. High levels of confinement lead to cell sorting while reducing matrix confinement triggers the collective fluidization of cell motion. Cell sorting, which depends on cell-cell adhesion, is crucial to this phenomenon. Burst-like migration does not occur for spheroids that have not undergone sorting, regardless of the degree of matrix confinement. Using computational Self-Propelled Voronoi modeling, we show that spheroid sorting and invasion into the matrix depend on the balance between cell-generated forces and matrix resistance. The findings support a model where matrix confinement modulates 3D spheroid sorting and unjamming in an adhesion-dependent manner, providing insights into the mechanisms of cell sorting and migration in the primary tumor and toward distant metastatic sites.

Minimal Morphoelastic Models of Solid Tumour Spheroids: A Tutorial

Tumour spheroids have been the focus of a variety of mathematical models, ranging from Greenspan's classical study of the 1970 s through to contemporary agent-based models. Of the many factors that regulate spheroid growth, mechanical effects are perhaps some of the least studied, both theoretically and experimentally, though experimental enquiry has established their significance to tumour growth dynamics. In this tutorial, we formulate a hierarchy of mathematical models of increasing complexity to explore the role of mechanics in spheroid growth, all the while seeking to retain desirable simplicity and analytical tractability. Beginning with the theory of morphoelasticity, which combines solid mechanics and growth, we successively refine our assumptions to develop a somewhat minimal model of mechanically regulated spheroid growth that is free from many unphysical and undesirable behaviours. In doing so, we will see how iterating upon simple models can provide rigorous guarantees of emergent behaviour, which are often precluded by existing, more complex modelling approaches. Perhaps surprisingly, we also demonstrate that the final model considered in this tutorial agrees favourably with classical experimental results, highlighting the potential for simple models to provide mechanistic insight whilst also serving as mathematical examples.

Stress Relaxation-Induced Colon Tumor Multicellular Spheroid Culture Based on Biomimetic Hydrogel for Nanoenzyme Ferroptosis Sensitization Evaluation

Ferroptosis has recently become a research hotspot, and the induction of tumor cell ferroptosis has emerged as a powerful method for tumor therapy. However, the efficiency of tumor cell ferroptosis induction remains unmet for clinical use, which may be attributed to the large discrepancies between in vitro and in vivo models. To address this issue, in this study, a hydrogel platform with stress relaxation is utilized to develop a multicellular spheroid model of the DLD1 colon cancer cell line through cancer cell self-organization. The spheroids are highly similar to real tumor tissue, and ferroptosis resistance at the transcriptional, protein, and cellular levels. Collaboration of the ferroptosis induction reagent erastin and the nanoenzyme MnZnFe₂ O₄ @PEG-COOH to overcome the ferroptosis resistance of the spheroids is also demonstrated. Taken together, this study demonstrates the effectiveness of the model developed using this hydrogel platform for further mechanistic studies, and for the assessment of novel cancer treatment strategies based on ferroptosis.

Two-phase model of compressive stress induced on a surrounding hyperelastic medium by an expanding tumour

In vitro experiments in which tumour cells are seeded in a gelatinous medium, or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally-supplied, diffusible nutrient (e.g., oxygen), affect the tumour's growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour's growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour's growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour's equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination.

Modelling glioma progression, mass effect and intracranial pressure in patient anatomy

Increased intracranial pressure is the source of most critical symptoms in patients with glioma, and often the main cause of death. Clinical interventions could benefit from non-invasive estimates of the pressure distribution in the patient's parenchyma provided by computational models. However, existing glioma models do not simulate the pressure distribution and they rely on a large number of model parameters, which complicates their calibration from available patient data. Here we present a novel model for glioma growth, pressure distribution and corresponding brain deformation. The distinct feature of our approach is that the pressure is directly derived from tumour dynamics and patient-specific anatomy, providing non-invasive insights into the patient's state. The model predictions allow estimation of critical conditions such as intracranial hypertension, brain midline shift or neurological and cognitive impairments. A diffuse-domain formalism is employed to allow for efficient numerical implementation of the model in the patient-specific brain anatomy. The model is tested on synthetic and clinical cases. To facilitate clinical deployment, a high-performance computing implementation of the model has been publicly released.