Tissue-Intrinsic Shape Mechanics in Growing Pre-Migratory Tumor Spheroids

TL;DR

This work addresses how tissue-intrinsic mechanical interactions steer the early morphogenesis of growing pre-migratory tumor spheroids. It introduces a 3D Graph Vertex Model with a novel, graph-based cell-division algorithm to simulate proliferating spheroids featuring a surface, living-necrotic interface, and necrotic core, decoupled from ECM effects. The study identifies a triad of interfacial tensions, growth heterogeneity, and tissue rheology as the key determinants of smooth versus lobulated morphologies, with differential proliferation and active fluctuations modulating shape evolution and relaxation. The results imply mechanical instabilities can underpin early invasive behavior and establish a scalable framework for exploring fully 3D tissue mechanics, paving the way to incorporate ECM mechanics in future work. Overall, the Graph Vertex Model with division provides a versatile toolkit for understanding how physical constraints shape tumor morphology during growth and potential metastatic progression.

Abstract

One of the hallmarks of pre-migratory tumors is the progressive loss of compact morphology. To investigate how tumors may intrinsically regulate their shape during growth, we employ a three-dimensional (3D) vertex model of multicellular aggregates that incorporates key structural features of tumor spheroids, including its surface, a proliferative rim, and a necrotic core. Focusing exclusively on tumor-intrinsic mechanical interactions, we examine how their collective effects guide morphological evolution en route to metastasis. We show that spheroids acquire lobulated morphologies through an interplay between differential tensions at the spheroid surface and the living-necrotic interface (LNI), together with differential growth within the proliferative rim. In addition, spheroid shapes can be substantially modulated by tissue rheological properties emerging from active, cell-scale forces. Our cell- and tissue-scale simulations of tumor morphologies are enabled by a computational framework that overcomes a major limitation of 3D vertex models - the lack of cell-division - by introducing a graph-based polyhedral-division algorithm within the Graph Vertex Model (GVM).

Figures (4)

• Figure 1: Mechanical model of smooth and lobulated tumor growth.a (top) MR image of a WHO grade I meningioma displaying a regular shape. (bottom) MR image of a WHO grade III meningioma displaying an irregular shape. Shape is quantified by surface factor, defined as ${\rm SF}=A_{\rm{sphere}}/A$, where $A$ denotes the surface area of the organoid and $A_{\rm sphere}$ the surface area of a sphere with the same volume. b The metagraph of GVM. On the left, typical vertex, edge, polygon and cell in an aggregate are shaded in blue, cyan, magenta and yellow color, respectively. On the right, the metagraph of GVM is depicted using same color code for nodes which are connected by directed arrows indicating the underlying hierarchy. Contextual properties or signs of vertex $\rightarrow$ edge, edge $\rightarrow$ polygon and polygon $\rightarrow$ cell relationships are indicated by $s$, $\sigma$ and $\Sigma$, respectively. c The two different proliferation profiles (exponential and step distribution) illustrated schematically. d, e Tumor growth under nutrient-limited conditions and different interfacial tensions between live and necrotic layers. Snapshots show tumor spheroids and their cross-sections, growing from $N = 50$ to $N = 2000$ cells. Inset: A cell before and after cell division, highlighted in red. The cell grows twice its regular size and divides into two daughter cells. The more spherical spheroid is obtained at $\Gamma=1$, $\Gamma_{\rm LNI} = 9$, and $\lambda = 1$ (panel d), whereas the more lobulated one is obtained at $\Gamma=1$, $\Gamma_{\rm LNI} = 1/9$, and $\lambda= 2$ (panel e). In the cross-sectional view, the dark gray polygons represent necrotic cells and the light red and magenta polygons represent live cells (red polygons represent the first live layer and magenta polygons represent the second live layer).
• Figure 2: Geometry, topology and the graph transformation for a 3D cell division.a The left and right columns show cells before and after cell division, respectively. Cell $c_1$ divides into two daughter cells, $c_1$ and $c_2$. The cleavage plane is highlighted in dark blue, and new or intersected polygons are marked red. The dark red polygon is shared with cell $c_3$ before division. Polygons above and below the red polygons are white and gray, respectively. b Polygons of $c_1$ are unwrapped and projected onto 2D before (left) and after (right) division, with the cleavage plane appearing as a straight blue line. Non-$c_1$ polygons are shown with dashed lines. c Subgraphs in the left and right columns highlight the components involved in division from the perspective of the dark red polygon. All nodes follow the same color code as in panel b. The middle column shows the graph transformation for cell division, with deleted and newly created relationships in red and green color, respectively. Transformation of contextual properties of new relationships are compactly shown at the bottom of the middle column.
• Figure 3: Morphology of simulated tumor spheroids as a function of differential tensions and the thickness of the proliferative rim.a, b The reduced volume $v$ and reduced perimeter $S$ versus the number of live layers $\lambda$ for $\Gamma=1$ and $\Gamma_{\rm LNI} \in \{ 1/9, 1/3, 1, 3, 9 \}$ (blue, purple, pink, red, and orange curves, respectively). The error bars denote the standard deviation across independent simulation instances. c-e 2D spheroid cross-sections corresponding to final states with 2000 cells. The necrotic cells are represented by gray polygons, the outermost live layer of cells is represented by red polygons, the second live layer is represented by magenta polygons, the third by blue, and the fourth by cyan. The cases shown include $\Gamma_{\rm LNI} = 9$ with $\lambda \in \{ 1, 2, 4 \}$ (c), $\Gamma_{\rm LNI} = 1/9$ with $\lambda \in \{ 1, 2, 4 \}$ (d), and $\Gamma_{\rm LNI} = 1/9$ with $\lambda = 2$ and $\Gamma \in \{ 1, 2 \}$ (e). Thick black and red outlines highlight the comparative shape of LNI and spheroid surface as a consequence of $\Gamma_{\rm LNI}$ and $\Gamma$. f Bar plot of the reduced volume $v$ and the reduced perimeter $S$ for the spheroid surface tensions $\Gamma \in \{0.5, 1, 2\}$ at two different sets of parameters: $\Gamma_{\rm LNI} = 1/9$, $\lambda = 2$ (most irregular tumor case at $\Gamma = 1$) and $\Gamma_{\rm LNI} = 9$, $\lambda = 1$ (least irregular tumor case at $\Gamma = 1$). g, h The reduced volume $v$ and reduced perimeter $S$ versus the number of live layers $\lambda$ for $\Gamma=1/2$ and $\Gamma_{\rm LNI} \in \{ 1/9, 1, 9 \}$ (blue, pink, and orange curves, respectively). Insets in panels a and g show representative final simulation snapshots. i 2D spheroid cross-sections corresponding to cases where $\Gamma = 1/2$, shown for $\Gamma_{\rm LNI} = 1/9$ and $\lambda = 1$ as well as $\Gamma_{\rm LNI} = 9$ and $\lambda \in \{ 1, 4 \}$.
• Figure 4: Effect of nutrient distribution profile and amplitude of active fluctuations on tumor morphology.a Reduced volume $v$ as a function of the proliferative-rim thickness $\lambda$, for $\Gamma=1$ and $\Gamma_{\rm LNI} \in \{ 1/9, 1, 9 \}$ (blue, pink, and orange curves, respectively). Solid and dashed curves show the results from simulations with exponential and step distibution of nutrients, respectively. The error bars denote the standard deviation across independent simulation instances. b, c 2D spheroid cross-sections corresponding to final states with 2000 cells at $\Gamma_{\rm LNI}=1/9$, $\lambda=2$ and $\Gamma_{\rm LNI}=9$, $\lambda=4$, comparing the results from exponential and step proliferation profile (left and right, respectively). d Dependence of the reduced volume $v$ on the amplitude of active tension fluctuations $\sigma$ for two distinct parameter sets with the most extreme tumor shapes previously observed at $\sigma = 0$ and $\Gamma = 1$ ($\Gamma_{\rm LNI} = 1/9$, $\lambda = 2$ and $\Gamma_{\rm LNI} = 9$, $\lambda = 1$). Insets show representative tumor morphologies. e 2D spheroid cross-sections corresponding to final states with 2000 cells at $\Gamma_{\rm LNI}=1/9$, $\lambda=2$ comparing most extreme tension-fluctuations amplitudes, $\sigma=0$ and $\sigma=0.5$ (left and right, respectively). f Fractions $f$ of surface polygon side numbers $n$ at $\Gamma_{\rm LNI} = 9$, $\lambda = 1$ (left) and $\Gamma_{\rm LNI} = 1/9$, $\lambda = 2$ (right) for $\sigma\in \{ 0, 0.2, 0.5 \}$ (green, red, and blue bars, respectively). There is an increase of hexagonal surface polygons near the optimal amplitude of edge–force fluctuations $\sigma \approx 0.2$.