Computational model of cancer invasion

To study the collective invasion of solid tumors, we developed an in silico model using the software CompuCell3D (CompuCell3D.org) version 4.6.0 [34], a modeling environment implementing the lattice-based Cellular Potts Model (CPM) framework [32,33,35]. The CPM is a stochastic, energy-minimizing framework for simulating multicellular systems in which cells are represented as extended, deformable domains on a discrete lattice. The mathematical formulation of the CPM energy functional and the Metropolis algorithm governing system dynamics are presented below. The total energy of the system is given by:

(1)

The first term captures cell–cell and cell–medium interactions through a contact energy matrix , where denotes the type of the cell occupying lattice site . The Kronecker delta function ensures that this energy contribution only arises at interfaces between distinct cells, thus quantifying interfacial tension due to differential adhesion. Lower values of J correspond to stronger adhesion, while higher values indicate weaker adhesive interactions. The second term imposes a volume constraint on each cell. is the current volume of cell , and is its prescribed target volume. Deviations from the target are penalized with strength , thereby enforcing volume conservation and preventing biologically unrealistic cell expansion or shrinkage. This constraint is crucial in capturing the physical limitation of space and mechanical pressure within dense tumor tissues. The third term encodes the migratory responsiveness of leader cells. is the local chemoattractant concentration at position , and is the migratory sensitivity coefficient that quantifies how strongly leader cells respond to gradients in the chemical field. The Dirac delta function restricts this term to only those lattice sites occupied by leader cells, enabling directed migration along chemoattractant gradients. In this model, follower cells are non-migratory and experience only passive motion through adhesion and growth [12].

Cell configurations evolve stochastically through attempted lattice updates using the Metropolis algorithm, where each attempted cell ID copy from site to a neighboring site is accepted with a probability governed by the Boltzmann distribution:

(2)

where is the change in energy associated with a proposed lattice update, and T is the effective simulation temperature controlling membrane fluctuations or biological noise. One Monte Carlo Step (MCS) comprises a number of attempted lattice-site copy events equal to the total number of lattice sites, so that each lattice site is selected once on average. Simulations were run for 700 MCS, corresponding to approximately 36 h of biological time. This framework was implemented using CompuCell3D (version 4.6.0), a modular simulation environment for CPM that incorporates PDE solvers, cell tracking modules, and customizable cell behaviors, such as chemotaxis and volume regulation [34].

Tumor domain, spatiotemporal calibration, and simulation parameters

The computational tumor represents key features of non-small cell lung cancer (NSCLC) tumor spheroids used in experimental systems such as the Spatiotemporal Genomic and Cellular Analysis (SaGA) platform [12].

This simplified model represents an idealized in vitro invasion assay without explicit representation of a basement membrane, organ tissue context, or stromal microenvironment elements present in vivo, but secondary to the mechanical cell-cell interactions that are the focus of this study. While tumor invasion typically begins as 3D spheroids embedded in extracellular matrix (ECM), we adopted a quasi-two-dimensional cross-section abstraction representative of experimental invasion assays used in experimental oncology, where spheroids are embedded in Matrigel or collagen, discretized on a lattice () with periodic boundary conditions along the horizontal (x) direction, as illustrated in Fig 1. This domain size was chosen to accommodate invasive fronts extending up to 500 micrometers from the initial tumor while maintaining computational tractability for large-scale parameter sweeps.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. Geometry abstraction from experimental spheroids to in silico domains.

(A) Experimental tumor spheroid embedded in ECM with leader-driven finger-like protrusions at the invasive front. (B) In silico spheroid under a radial migratory field; dashed concentric rings indicate radial distances, and the “Radial Gradient” annotation marks outward bias. Leader cells (blue) respond chemotactically to the gradient and drive directional invasion, while follower cells (green) proliferate and maintain tumor integrity through cell-cell adhesion. (C) Rectangular slab abstraction used in this study. Cells are initialized at the base with leader cells (blue, 25% of population) possessing a high migration coefficient () and follower cells (green, 75% of population) exhibiting a high proliferative probability (PP). All cells are initialized at the base and exposed to a vertical stimulus that increases along the y-axis (arrow), with only leader cells responding through directed migration while follower cells move passively through mechanical coupling to the leaders. This reduced 2D geometry preserves directional invasion behavior and leader-follower dynamics while enabling reproducible, high-throughput parameter sweeps. S1 Video illustrates the temporal evolution of the corresponding 3D simulation, confirming that the reduced 2D slab abstraction effectively captures the key structural features of tumor invasion.

https://doi.org/10.1371/journal.pcbi.1014747.g001

In adopting this 2D abstraction, we made three simplifying assumptions. First, 2D simulations preserve key 3D invasion dynamics: prior CPM studies [36–38] have demonstrated that 2D cross-sections retain essential morphological features, including finger formation, growth dynamics, and spatial patterning, in 3D tumor models. Our preliminary 3D spheroid simulations (S1 Fig, S1 Video) confirmed that invasive area, finger counts, and cluster incidence differed by less than 15% between 2D and 3D under equivalent parameter sets. Second, ECM remodeling does not qualitatively alter invasion modes within the simulation timeframe. We have previously quantified collagen fiber alignment as a consequence of cancer cell invasion over 24-hour periods [39,40], finding that significant matrix reorganization occurs on timescales of 24–48 hours. Since our simulations span approximately 36 hours (700 MCS), the static ECM assumption is reasonable for capturing initial invasion dynamics before substantial matrix feedback becomes dominant. Third, the tumor-front curvature inherent to spheroid geometry does not fundamentally alter the parameter dependence of invasion phenotypes. While absolute metric values may differ between planar and curved fronts, the transitions between no invasion, single-cell, bulk collective, and multimodal invasion are governed primarily by the balance between adhesion and motility, which is preserved in the 2D slab geometry.

The model consisted of two biologically distinct cell types: leader cells (LCs), which exhibit strong directed migration, and follower cells (FCs), which proliferate stochastically but lack intrinsic motility. We initialized the simulated tumor as a planar slab of follower cells (FCs) occupying the bottom of the domain, spanning the full width of 500 pixels and a height of 21 pixels (y = 0–21), the latter corresponding to approximately 52.5 µm. This configuration contained approximately 1167 follower cells, with exact counts varying slightly due to stochastic initialization. A random subset of 25% of these cells was then reassigned as leader cells (LCs), consistent with experimental observations of leader fractions of approximately 30% in NSCLC spheroids [41]. To confirm that this choice does not qualitatively alter the findings, simulations were additionally performed at leader fractions of 15% and 35%, bracketing the chosen value. The spatial structure of all five invasion metric landscapes and their parameter sensitivities were preserved across all three conditions (S9 Fig), supporting the robustness of the 25% condition as a representative value. This configuration mimics the basal cross-section of a spheroid tumor embedded in the ECM, thereby maintaining the invasion polarity and architecture observed in vitro.

A spatially static, non-diffusing chemoattractant gradient field (implemented as a CompuCell3D chemical species named ’MV’ for migration vector) was imposed along the y-axis to simulate a constant directional migration cue. The gradient was defined as c(x,y) = y/g, where g = 1 sets the gradient strength. This linear gradient represents a fixed directional stimulus consistent with vertical invasion observed in Matrigel-based assays [42], and could correspond biologically to oxygen gradients, growth factor gradients, or other chemotactic cues that guide leader cell migration away from the tumor core. This quasi-2D slab configuration preserves essential directional invasion dynamics while reducing computational complexity, enabling systematic exploration of key biophysical parameters across more than 13,000 simulations.

The correspondence between simulation spatiotemporal scales and in vitro tumor cell behavior was established by calibrating leader cell migration speeds to experimentally measured values from 3D collagen-based invasion assays [12]. Leader cells exhibited mean velocities of (1.1 0.3) µm/min. We measured leader cell displacement at (intermediate motility) and obtained 0.4 px/MCS. Setting the spatial scale to 1 px 2.5 µm (guided by typical carcinoma cell diameters of 15 µm to 20 µm) and the temporal scale to 1 MCS 3.1 min, we achieved simulated leader velocities of approximately 0.32 µm/min at and 0.9 µm/min to 1.2 µm/min at , matching experimental measurements. Our standard simulation duration of 700 MCS represents approximately 36 hours of biological time, consistent with SaGA-based invasion imaging [12,18].

Volume constraints were imposed on all cells with a target volume V₀ = 10 lattice units and volume constraint strength . The volume constraint strength was chosen to allow realistic cell shape fluctuations while preventing unrealistic cell fragmentation or excessive volume deviation, following previous CPM tumor models [36]. The main simulation study examined the impact of three biophysical control parameters on tumor invasion dynamics. The first parameter, leader–follower contact energy (), regulated the adhesive interaction strength between LCs and FCs and was varied across a range of to +5 in 11 discrete levels. In the CPM, higher contact energy () corresponds to weaker adhesion between cell types, while lower (more negative) indicates stronger adhesive interactions. The second parameter, the migration coefficient (), modulated the response of LCs to the chemoattractant gradient, ∇. We varied λ from 0 to 30 in 11 discrete levels, it represents the migratory sensitivity and can be interpreted as the effective force per unit concentration gradient experienced by leader cells. The third parameter, proliferative probability (PP), represented the fraction of FCs that are proliferation-competent, with values ranging from 0 to 1.0 in 11 discrete levels.

These parameter ranges were identified through preliminary exploration spanning the full spectrum of qualitatively distinct invasion phenotypes. Preliminary simulations (n = 50 across broader ranges: , , ) revealed that parameter values outside the selected ranges produced either uniform behavior or computational artifacts. Specifically, resulted in uniformly static, non-invasive tumors due to excessive cell–cell cohesion, while caused complete tumor fragmentation into isolated cells. Similarly, induced boundary artifacts due to excessively rapid cell displacement, exceeding the lattice resolution. At the lower end, tumors remained compact with minimal morphological variation. The refined ranges (, ) were chosen to span regimes capable of producing all four invasion modes (no invasion, single-cell, bulk collective, multimodal).

Each of the 11³ = 1331 unique parameter combinations was replicated 10 times using unique random seeds (0–9), yielding 13 310 total simulations. Follower cell proliferation was implemented using a clock-based system: at the start of the simulation, a subset of size (where is the initial follower count) was designated as proliferation-competent. Proliferation-competent cells underwent mitosis when their volume exceeded 20 pixels (twice the target volume) and their internal clock surpassed a division time drawn uniformly from 75 ± 50 MCS, i.e., MCS,corresponding to a mean inter-division time of . This accelerated proliferation rate reflects rapid growth in aggressive cancer cell lines cultured in 3D ECM [43]. Upon division, cells were bisected along random axes, and daughter cells inherited a proliferation-competent state with reset timers. Leader cells were not assigned proliferation capacity, consistent with experimental observations that highly motile, invasive leader cells often exhibit reduced proliferation compared to followers [12]. Parameter values were motivated by experimental estimates from biological observations on collective cancer invasion [3,12]. A complete summary of all simulation parameters, including dimensions, fixed contact energies, and control parameters, is provided in Table 1.

Download:

• PNG
  larger image
• TIFF
  original image

Table 1. Simulation and biophysical parameters used in the leader–follower tumor invasion model. Dimensions are expressed in terms of L (length), T (time), and E (energy). Parameters marked with an asterisk (^*) were systematically varied during the parametric scan. Superscript notation indicates: (i) calibrated to experimental data, (ii) selected from preliminary parametric exploration, (iii) based on literature values or experimental observations, (iv) standard CPM framework parameters.

https://doi.org/10.1371/journal.pcbi.1014747.t001

Computational implementation and code availability

Post-processing used NumPy 1.21, SciPy 1.7, NetworkX 2.6, and Matplotlib 3.4. Each simulation ran for 700 MCS with periodic boundary conditions along the x-axis, requiring approximately 12–18 minutes per replicate on an Intel Xeon E5-2680 processor (2.4 GHz, 8 cores). The complete parameter sweep required approximately 2,800 CPU-hours, distributed across a 48-core computing cluster. Simulations were executed in batches using environment variables to specify parameter combinations (J_LF, MU, PP) and replicate indices (REP), enabling efficient parallelization. All simulation code, post-processing scripts, and analysis notebooks are available in the GitHub repository: https://github.com/Jiang-Lab/Tumor_Invasion_Model.

Invasion metrics and phenotype classification

For each simulation, spatial and topological features were extracted at the final time point (MCS = 700) using a combination of breadth-first search (BFS), neighborhood analysis, morphological processing, and cell-type–specific annotation. The main tumor mass was identified by constructing a contact graph in which each cell is represented as a node and cell–cell contacts define edges. Starting from all cells at the domain base (y = 1), we performed a BFS to identify all physically connected cells, thereby defining the set of main tumor cells. Cells were considered neighbors if they shared at least one lattice edge (von Neumann neighborhood). This graph-based approach naturally captures the connected component corresponding to the primary tumor while excluding spatially detached clusters and solitary defectors.

The invasive area was defined as the cumulative area occupied by the main tumor mass [18], computed by summing the volumes of all cells belonging to the connected component anchored at y = 1. This metric reflects cohesive tumor expansion and requires cells to maintain physical contact with the core structure. The invasive front boundary is shown by the red curve in Fig 2. The infiltrative area was computed as the area of the convex hull enclosing all tumor cells (both main tumor and detached), capturing the overall spatial footprint of the tumor, including dispersed single cells and clusters [44]. This metric provides a global measure of spatial dispersion, irrespective of physical connectivity, and is represented by the orange outermost boundary in Fig 2.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. Distinct emergent invasion phenotypes in leader–follower tumor simulations.

Representative snapshots of tumor morphologies at endpoint (MCS = 700) for the four canonical invasion phenotypes, each arising from unique combinations of leader motility, follower proliferation, and intercellular adhesion parameters. (A) No Invasion: Tumor remains compact with no detachment or dispersal beyond initial bounds. Invasive and infiltrative areas are nearly identical; fingers, defectors, and clusters are absent. (B) Single-Cell Invasion: Individual cells (cyan stars) detach from the tumor mass, increasing the infiltrative area without forming collective structures. (C) Bulk Invasion: Finger-like protrusions (yellow pentagons with navy edges) emerge, indicative of coordinated collective migration; defectors and clusters are not observed. The invasive area is large and contiguous. Fingers are defined as protrusions pixels (30 µm) in the y-direction, separation of at least 20 pixels ( 60 µm) from neighboring peaks in the x-direction, and width at half-height prominence pixels. (D) Multimodal Invasion: The coexistence of collective fingers, solitary defectors, and detached clusters (magenta Xs) reflects a heterogeneous expansion mode that combines cohesive and dispersed strategies. Standard annotations across panels: core boundary (red), outermost boundary (orange), lowest point (purple dashed), follower cells (light green circles), leader cells (blue arrows), defector cells (cyan stars), cluster centroids (magenta X), boundary peaks (yellow pentagons with navy edges).

https://doi.org/10.1371/journal.pcbi.1014747.g002

To characterize morphological complexity, we extracted geometric features from the simulated tumor front. For each x-coordinate, we scanned vertically from the top of the domain (y = 299) downward to identify the highest y-position occupied by a main tumor cell, yielding the tumor boundary profile y_tumor(x). This one-dimensional boundary was interpolated using cubic smoothing splines (SciPy) to facilitate derivative calculations and peak detection. Invasive fingers were detected using a peak-finding algorithm applied to the interpolated boundary profile. Peaks were classified as fingers if they satisfied three criteria; prominence pixels in the y-direction (corresponding to 30 µm) protrusion height), separation of at least 20 pixels ( 60 µm) from neighboring peaks in the x-direction, and width at half-prominence pixels. These thresholds were chosen to identify biologically meaningful finger-like protrusions while filtering spurious peaks arising from single-cell fluctuations, following established definitions in prior computational studies [45,46]. To further reduce over-counting of closely spaced peaks, we applied a post-processing merge step: peaks separated by less than 15 pixels were consolidated into a single finger, retaining only the leftmost peak. The resulting count of merged peaks constituted the final finger metric.

Singles were defined as individual cells (leader or follower) with no contact neighbors. Specifically, a cell was classified as a single defector if it shared no lattice edges with any other tumor cell (von Neumann neighborhood criterion) and was located above the minimum y-coordinate of the main tumor. This definition captures solitary cells that have fully detached and migrated away from any collective structure. Detached clusters were operationally defined as spatially separated groups of two or more tumor cells that maintained internal connectivity but were entirely disconnected from the main tumor component [47]. Clusters were identified using BFS on all cells not belonging to the main tumor: for each follower cell outside the main tumor, we initiated a BFS to identify its connected component. If this component contained two or more cells and had no contact with the main tumor, it was classified as a detached cluster. Cluster composition was quantified by counting the number of leaders and followers within each cluster, enabling classification of purely follower-based aggregates, leader-only satellites, and mixed clusters. Cluster centroids were computed as the mean x- and y-coordinates of member cells. Cluster persistence time was measured from emergence to either dissolution (loss of all intercellular connections within the cluster or complete reintegration into the main tumor mass) or right-censoring at the simulation endpoint (t = 700 MCS). Clusters persisting until the final time point were recorded separately to distinguish early-forming clusters with complete dissolution histories from late-forming clusters constrained by the finite simulation window. The majority of the clusters lived through the end of the simulation; therefore, longer persistence time corresponds to earlier cluster formation.

Representative examples of the annotated morphological features associated with each invasion phenotype are shown in Fig 2. Each simulation was assigned a phenotypic label using a rule-based classification framework designed to identify biologically interpretable modes of tumor invasion. The classification used endpoint invasion metrics measured at MCS = 700, including invasive area, infiltrative area, finger-like protrusions, solitary defectors, and detached clusters. Importantly, invasion was defined operationally by a morphological transition from a smooth, compact tumor boundary to a boundary displaying protrusions, cellular detachment, or dispersed infiltrative structures, rather than by a fixed distance threshold from the tumor core. Using these criteria, simulations were categorized into four phenotypes: No Invasion, Single-Cell Invasion, Bulk Invasion, and Multimodal Invasion. This scheme was designed to parallel experimental classifications of tumor invasion behavior [5].

Simulations were classified as No Invasion (Fig 2A) when the invasive and infiltrative areas were nearly identical, and no fingers, solitary defectors, or detached clusters were detected, indicating that the tumor remained morphologically compact and did not substantially breach its initial boundary. The Single-Cell Invasion phenotype (Fig 2B) was assigned when the infiltrative area exceeded the invasive area and solitary defectors were present, while finger-like protrusions and detached clusters were absent. This phenotype represents dispersal of individual cells without coordinated collective invasion. Simulations were classified as Bulk Invasion (Fig 2C) when they exhibited a large invasive area with finger-like protrusions but no solitary defectors or detached clusters, indicating coherent, strand-like collective expansion of the tumor front. Finally, the Multimodal Invasion phenotype (Fig 2D) was assigned when cohesive invasive structures, such as fingers, coexisted with dispersed elements, such as solitary defectors or detached clusters, and the infiltrative area substantially exceeded the invasive area. This phenotype captures a hybrid invasion mode in which collective and individual invasion occur within the same spatial domain.

This automated classification pipeline enabled the high-throughput categorization of invasion phenotypes across the whole parameter space of 13 310 simulations. We saved the outputs in CSV format and compiled them into aggregate data tables for phenotype classification. The robustness of classifier thresholds was confirmed by varying finger-prominence and invasive-to-infiltrative area ratio criteria by from baseline values; phenotype boundaries and classifications remained stable across the parameter space.

Distinct invasion metric sensitivities to adhesion, motility, and proliferation

To dissect how cellular biophysical properties shape emergent tumor invasion dynamics, we systematically varied leader–follower contact energy (), leader migration coefficient (), and follower proliferative probability (PP) across a three-dimensional parameter space encompassing 13 310 simulations (1,331 unique parameter combinations with n = 10 replicates each). Each simulation ran for 700 MCS (approximately 36 h of biological time), and we quantified five invasion metrics at endpoint: invasive area (expansion of the connected tumor mass), infiltrative area (total spatial footprint including dispersed cells), number of finger-like protrusions, single invading cells (singles), and detached clusters. These metrics capture distinct aspects of tumor spatial organization and collectively define invasion phenotypes.

Invasive area, measuring the expansion of the connected tumor mass, was maximized under intermediate adhesion ( to 2) and elevated migration () (S2 Fig). Strong adhesion significantly reduced invasive expansion compared to intermediate adhesion. Under strong adhesion conditions (), mean invasive area was 33 875 µm², representing a 1.9-fold reduction compared to intermediate adhesion which yielded 64 463 µm². Weak adhesion () further reduced invasive area to 41 856 µm², which is 1.5-fold lower than intermediate adhesion. Migration exhibited a substantially stronger effect on invasive expansion. High migration () produced a 4.2-fold increase in invasive area compared to low migration (). Specifically, high migration conditions generated mean invasive area of 76 125 µm² whereas low migration yielded only 18 163 µm². Strong adhesion prevented structural remodeling, while very weak adhesion reduced cohesion, both of which limited expansion. Higher PP increases follower cell mass within the tumor, whereas adhesion and leader motility primarily determine invasion mode.

In contrast, the infiltrative area, which captures the full spatial footprint, including dispersed single cells and clusters, was dictated primarily by migration. The Infiltrative area increased monotonically with the migration coefficient and exhibited an 8.5-fold increase from low to high migration conditions. Low migration () produced mean infiltrative area of 22 369 µm² while high migration () generated 191 200 µm²(S3 Fig). Adhesion also modulated infiltrative spread under high migration conditions. Weak adhesion () increased infiltrative area to 183 406 µm², representing a 1.7-fold increase compared to intermediate adhesion. Thus, directional motility alone is sufficient for extensive tumor spread, although it does not necessarily lead to cohesive growth.

Finger counts peaked at intermediate adhesion ( to 2), yielding a marginal mean of 7.11 fingers (averaged across all migration coefficients and proliferative probabilities), and at strong migration () with a marginal mean of 9.74 fingers (averaged across all adhesion strengths and proliferative probabilities) (S5 Fig). Strong adhesion () restricted protrusive branching (mean = 4.03 fingers), while weak adhesion () fragmented collectives into singles or clusters (mean = 6.11 fingers). This non-monotonic dependence on adhesion is consistent with experimental studies showing that optimal intercellular adhesion promotes dynamic collective invasion [45,46]. Proliferation had a negligible influence on finger formation (Pearson , p = 0.985). The number of singles increased sharply with and weak adhesion (), reflecting a transition toward mesenchymal-like escape (S4 Fig). Weak adhesion produced mean = 175.79 singles compared to 1.53 under strong adhesion and 53.06 under intermediate adhesion. While such single cells enable early dissemination, clinical evidence suggests that solitary circulating tumor cells frequently enter dormancy or undergo apoptosis, thereby limiting their metastatic potential [48].

Detached clusters, defined as cohesive aggregates that separate from the core, formed in 28.3% of all simulations (3 763 out of 13 310) and were most prevalent under intermediate-to-weak adhesion ( to 3) and high migration () (S6 Fig). Intermediate adhesion produced a mean of 1.69 clusters per simulation, while weak adhesion yielded 1.78 clusters. High migration () resulted in 70.1% of simulations exhibiting clusters, with mean of 5.00 clusters when present. In this regime, adhesion is strong enough to preserve intercellular cohesion within the aggregate yet weak enough to allow detachment from the primary mass. Larger clusters tended to persist until the end of simulations (700 MCS), while smaller aggregates often dissolved earlier. Again, proliferation (PP) showed no consistent effect on cluster frequency or stability (Pearson , p = 0.815). These observations parallel experimental findings that cluster-based dissemination is highly metastatic [47,49].

Correlation-based sensitivity analysis (S7 Fig) corroborated these trends: migration showed the strongest correlations with invasive area (Pearson r = 0.70, Spearman ), infiltrative area (Pearson r = 0.69, Spearman ), and finger counts (Pearson r = 0.80, Spearman ). Contact energy exhibited moderate-to-strong correlations with singles (Pearson r = 0.67, Spearman ) and infiltrative area (Pearson r = 0.57, Spearman ). Proliferation exhibited weak correlation across all metrics (Pearson and Spearman |r| < 0.03, all p > 0.05), reinforcing its limited role in determining invasion mode. These metric sensitivities were qualitatively preserved across simulations run at leader fractions of 15% and 35%, confirming that the reported parameter dependencies are not specific to the 25% baseline condition (S9 Fig).

To directly examine the marginal effects of proliferative probability on invasion metrics, we systematically compared metric-parameter relationships across all three parameters (Fig 3A-3E). Across all five invasion metrics, PP showed only minimal influence compared with the migration coefficient and contact energy. Increasing the migration coefficient () consistently elevated the invasive and infiltrative area, promoted the formation of fingers, and sharply increased the number of singles and detached clusters. Contact energy () exerted a non-monotonic effect, with peaks at intermediate adhesion (–2) that supported cluster formation and cohesive protrusions. By contrast, proliferative probability produced nearly flat curves, with only minor oscillatory deviations. While high proliferation increases follower availability and may create local crowding effects in the tumor core, the overall prevalence of invasion phenotypes and cluster frequency show no systematic dependence on PP. This suggests that mechanical parameters (adhesion and migration) are the primary determinants of invasion architecture. Thus, proliferation increased overall tumor bulk but did not enhance the spatial extension of the connected invasion front.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Migration and adhesion together determine invasion modes, while proliferation scales tumor bulk.

(A–E) Metric parameter relationships for invasive area (A), infiltrative area (B), fingers (C), singles (D), and detached clusters (E). The x-axis is a normalized parameter value (scaled to [0,1]). Original parameter value ranges were (migration coefficient) and (contact energy, lower values = stronger adhesion). Curves show the marginal dependence on proliferative probability(triangles, ), contact energy (circles, •), and migration coefficient (squares, ), all plotted as solid lines. Shaded bands denote mean standard error of the mean (SEM). Migration strongly increased invasive and infiltrative areas, singles, and clusters. Contact energy shows non-monotonic optima at intermediate adhesion, and proliferation induces minor changes. (F) Phenotype composition across proliferative probability (PP) bins, computed as normalized frequency counts across all 13,310 simulations, with error bars denoting mean s.d. across ten replicates per parameter combination. Phenotype proportions remain largely unchanged across PP bins, whereas migration () and adhesion () drive substantial phenotype transitions.

https://doi.org/10.1371/journal.pcbi.1014747.g003

To validate that these metrics capture consistent invasion features across geometries, we compared representative parameter combinations from each phenotype in 2D and 3D simulations: no invasion (), single-cell invasion (), bulk invasion (), and multimodal invasion (). Invasive area, finger counts, and cluster incidence differed by <15% between geometries across all phenotypes, confirming that the 2D domain preserves key invasion features across the parameter space explored. Together, these findings demonstrate leader cell motility and intercellular adhesion as the primary regulators of invasion architecture, with proliferation acting mainly as a scaling factor for tumor bulk. This functional decoupling underscores the central role of mechanical and migratory properties in shaping invasion phenotype and spatial spread.

Phenotypic phase-space mapping reveals prevalence of multimodal invasion

We classify emergent invasion behaviors within our leader-follower framework into four biologically interpretable phenotypes: No Invasion, Single-Cell Invasion, Bulk Invasion, and Multimodal Invasion. Phenotype labels were assigned by a rule-based classifier applied to multiple invasion metrics, including invasive area, infiltrative spread, finger counts, single cells, and detached clusters. The classification thresholds were grounded in biological principles: the infiltrative-to-invasive area ratio distinguishes phenotypes by definition; when these areas are nearly equal, no cellular dispersion occurs (no invasion or bulk invasion); when the infiltrative area substantially exceeds the invasive area, cells have dispersed as single cells or in a multimodal mix. Finger-prominence thresholds ( pixels height, pixels separation) follow established definitions [45,46]. To assess robustness, we spot-checked representative parameter combinations from each phenotype and varied thresholds by (e.g., finger height pixels). Phenotype classifications remained stable because phenotypes are defined by combining multiple criteria simultaneously, making them insensitive to minor adjustments in any single metric. This approach provides an objective mapping of complex invasion behaviors across the parameter landscape.

These categories align with the framework proposed by Friedl et al. [5], which organizes invasion behavior along a spectrum from single-cell migration to collective streaming. Our Single-Cell Invasion corresponds to single-cell escape, in which leader cells detach and migrate independently from the core tumor mass under weak adhesion. Bulk Invasion reflects collective streaming behavior, characterized by coordinated protrusions and preserved junctional integrity. No Invasion represents non-migratory, growth-dominant behavior. Finally, Multimodal Invasion shows the richest dynamics, with the simultaneous formation of collective fingers, single cells, and detached clusters, a phenotype increasingly observed in aggressive clinical tumors [49,50] but rarely captured by computational models. While our model captures these principal invasion modes through the mechanical balance of adhesion, motility, and proliferation, it represents a focused subset of the broader Friedl framework, abstracting away from features such as morphological plasticity, explicit multicellular polarity, and dynamic ECM remodeling that would require additional mechanistic complexity. Figs 4 and S8 illustrate how these phenotypes emerge over time.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. Temporal evolution of tumor morphologies across invasion phenotypes.

Rows show representative simulations for the four phenotypes, no invasion, single-cell invasion, bulk invasion, and multimodal invasion, while columns show time points (0, 350, and 700 MCS, corresponding to 0, 18, and 36 hours). Leader cells (red) respond to the vertical gradient, and follower cells (green) proliferate and maintain cohesion. In multimodal invasion, cyan represents detached clusters. No invasion (top row) remains compact, illustrating regimes of strong adhesion and negligible leader motility. Single-cell invasion (second row) emerges under weak adhesion and high leader motility, where leader cells detach and migrate individually. Bulk invasion (third row) features cohesive, finger-like protrusions that preserve junctional integrity, supported by high motility and proliferative replenishment. Multimodal invasion (bottom row) exhibits coexisting fingers, dispersed singles, and detached clusters, under intermediate adhesion and high leader motility. See S2 Video for full spatiotemporal dynamics.

https://doi.org/10.1371/journal.pcbi.1014747.g004

We visualized phenotype assignments in the 3D parameter space defined by leader–follower contact energy (), leader migratory coefficient (), and follower proliferative probability (PP). The resulting phase map (Fig 5A) reveals well-delineated phenotypic regions separated by sharp, nonlinear transitions as one or more control parameters vary. As summarized in S1 Table, the No Invasion phenotype (green) occupies approximately 22% of the parameter space primarily at low migration () and moderate to strong adhesion (), where leader cells either lack migratory drive or remain tightly bound to follower cells, preventing outward expansion. Single-Cell Invasion (blue) appears in a narrow but distinct band ( of parameter space) at weak adhesion () and low to intermediate migration (), where leaders detach and migrate independently with limited follower engagement. Bulk Collective Invasion (orange) emerges in approximately 23% of parameter space under strong adhesion (), moderate migration (), and moderate proliferation (0.1 < PP < 0.7), producing cohesive strands resembling the leader-follower chains observed in NSCLC spheroids [12]. In contrast, Multimodal Invasion (gold) spans a broad range ( of parameter space) of intermediate to weak adhesion (), high migration (), and moderate to high proliferation (PP > 0.3), representing the most prevalent invasion outcome in our simulation.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. 3D phase space of invasion phenotypes.

(A) Multidimensional phenotype map over leader–follower contact energy (), leader migration strength (), and follower proliferative probability (PP). Each voxel is assigned the most probable phenotype: green (No Invasion), blue (Single-Cell Invasion), orange (Bulk Invasion), or gold (Multimodal Invasion). Boundaries between regimes are sharp and nonlinear, and multimodal invasion occupies a large region characterized by moderate adhesion and strong migration across all PP. See S8 Fig for 2D cross-sections of the phase space. (B) The phenotype frequency distribution across 13 310 simulations. Multimodal invasion is the most prevalent outcome, with bulk and no invasion occurring under more restricted conditions, and single-cell invasion is comparatively rare. Error bars denote mean s.d. across 10 replicates. S3 Video shows the temporal progression, with early compact states giving way to increasing multimodal prevalence at later times.

https://doi.org/10.1371/journal.pcbi.1014747.g005

Notably, multimodal invasion is the most prevalent outcome in our simulations (Fig 5B), with more than half of all parameter combinations favoring this heterogeneous phenotype. This suggests that heterogeneous invasion, characterized by coexisting finger-like protrusions, detached clusters, and isolated individual cells, is a robust and recurrent tumor behavior. This phenotype emerges most frequently under intermediate adhesion and high motility, consistent with histopathological and experimental observations in invasive carcinomas, where collective strands often coexist with solitary migratory cells and multicellular clusters [10,49]. While specific tumor subtypes may differ in their parameter mappings, the phase structure suggests testable strategies for shifting tumors away from multimodal regions by jointly modulating adhesion and motility.

To examine the phenotypic impact of proliferation more directly, we assessed phenotype classification across the full parameter space. Across all simulations (13,310 total), the proportion of tumors in the bulk, multimodal, single-cell, and non-invasive classes remained essentially unchanged across proliferative probability (PP) bins. In sharp contrast, both and produced marked phenotype transitions. Statistical sensitivity analyses reinforced these observations: distance correlation analysis revealed strong dependencies of invasion metrics on migration (up to 0.79 for fingers) and moderate dependencies on adhesion (e.g., 0.68 for singles), while PP correlations were negligible () (S7 Fig). A multi-output random forest model trained on all three parameters (400 trees, mean-decrease-in-impurity criterion) showed a similar pattern: migration () and adhesion () accounted for virtually all explanatory power, while proliferation contributed negligibly (). Pearson and Spearman correlation coefficients similarly showed near-zero correlations between PP and all metrics (|r| < 0.03). Full statistical analysis methods, including distance correlation, random forest importance, and SEM computation, are provided in S1 Appendix. These results demonstrate that while proliferation affects the scale of tumor growth (reflected in a larger invasive area at high PP due to core expansion), it does not alter the phenotypic classification or the fundamental structure of invasion modes. Migration and adhesion remain the primary regulators of spatial architecture and invasion phenotype.

Cluster formation peaks under intermediate adhesion and strong migration

Multicellular clusters represent a distinct invasion mode capable of cohesive dissemination from the primary tumor mass. Clusters typically preserve leader–follower heterogeneity, wherein leader cells provide traction and directional persistence while followers maintain cohesion. Understanding the mechanical principles governing cluster formation in vitro provides a foundation for investigating multicellular escape strategies in more complex environments.

To examine the mechanistic drivers of cluster detachment, we systematically evaluated simulation outcomes and computed cluster-related metrics at every 10 MCS, including cluster counts, size distributions, cellular composition, and trajectory persistence, across 13 310 simulations. Cluster emergence depended strongly on the adhesion-motility balance. Heatmaps of cluster incidence (Fig 6A) showed a sharp increase under intermediate adhesion (–3) and strong motility (–30). The heatmap of total cells aggregated in clusters (Fig 6C) further revealed that this regime not only produces more cluster-forming events but also results in larger clusters: high-migration, intermediate-adhesion conditions yielded 30–65 cells in clusters on average, compared to fewer than 5 cells in low-migration or very-weak-adhesion regimes. These conditions define a mechanically permissive regime in which adhesion is low enough to permit detachment yet high enough to preserve intra-cluster cohesion, while high leader motility provides leader-driven traction for escape. By contrast, strong adhesion () or low motility () suppressed cluster formation, confining cells to the primary tumor mass.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 6. Cluster formation landscape: incidence, abundance, and composition across parameter space.

(A) Heatmap of cluster incidence (number of clusters per simulation) across adhesion (, y-axis) and migration (, x-axis). Each cell represents the mean count across 10 replicates. Blue indicates no or few clusters; red indicates high cluster frequency, peaking at intermediate adhesion (–2) and high migration (). (B) Histogram of cells per cluster showing the distribution of cluster sizes. Most clusters contained 4–8 cells, with rare aggregates exceeding 30 cells, with an average of 7 cells. (C) Heatmap of total cells per cluster (mean per simulation) showing that higher cluster incidence is associated with large aggregate sizes. Peak values (30–40 cells/cluster) occur in the same intermediate-adhesion, high-migration window. (D, E) Cluster composition analysis showing leader and follower cell counts across parameter space.

https://doi.org/10.1371/journal.pcbi.1014747.g006

Cluster composition showed substantial leader enrichment. Analysis across all parameter combinations revealed that leader fractions commonly reached approximately 60–70% in the cluster-forming regime, suggesting that leaders were consistently enriched within clusters, with a median of approximately 4 leaders per cluster versus 3 followers (Fig 6B). The leader fraction distribution (S13 Fig) exhibited consistent enrichment across proliferative probability values, peaking at approximately 70% in the multimodal invasion regime, indicating that clusters are not random aggregates but leader-dominated collectives with robust composition. This enrichment reflects the mechanistic role of leader cells in generating traction and directional persistence.

Across the explored range, PP exerted at most a secondary influence on cluster incidence. Varying PP altered the follower availability but did not qualitatively change the adhesion–motility trends or cluster stability. Fixing PP = 0.9 in parameter scans confirmed that proliferation primarily increased follower counts within clusters without eliminating leader dominance. This finding suggests that motility and adhesion, rather than proliferation, are the principal determinants of cluster detachment.

The aggregate distribution of cluster sizes was strongly right-skewed or heavy-tailed (Fig 7). Most clusters consisted of 4–8 cells, while rare aggregates exceeded 30 cells, with a mean cluster size of approximately 7 cells. Persistence time increased with cluster size (S14 and S15 Figs). Most clusters formed within of the simulation endpoint; because many persisted to 700 MCS, their recorded persistence times are right-censored. Panel insets (S11 Fig) highlight specific parameter combinations (e.g., ), where small clusters dominated, but rare, large aggregates persisted for the full duration of the simulation, reflecting heterogeneous outcomes characteristic of this regime.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 7. Representative cluster size distributions across migration and adhesion at fixed proliferation.

Shown for PP = 0.9 at a subset of combinations chosen to illustrate regimes with and without cluster formation. (See S11 Fig for the full panel.) Each panel shows the distribution of cluster sizes (y-axis) versus the number of clusters observed (x-axis), pooled across n = 10 replicates and sampled every 10 MCS. Bars are colored by persistence time, with lighter shades indicating longer-lived clusters. Persistence time is expressed in sampling units of 10 MCS, such that a value of 50 corresponds to 500 MCS of biological simulation time. Panels are arranged in the plane, with migration coefficient increasing from left to right and contact energy increasing from top to bottom, as indicated by the outer axis arrows. Panels labeled “No Clusters” indicate conditions under which no clusters formed. Across the representative conditions, strong-adhesion or low-motility regimes show few or no clusters, whereas intermediate adhesion (–2) combined with high motility () yields numerous small-to-intermediate clusters and occasional large, persistent aggregates, capturing the heterogeneous outcomes characteristic of this parameter window.

https://doi.org/10.1371/journal.pcbi.1014747.g007

Centroid displacement analyses showed that displacement increased monotonically with the migration coefficient () across the parameter space, with longer-persisting clusters consistently achieving greater total displacement (S12 Fig). At , for example, clusters with greater persistence times maintained sustained displacement over time, while shorter-lived clusters remained near their origin (S10 Fig). We additionally tracked splitting events, where one cluster fragmented into smaller aggregates or single cells, and merging events, where detached clusters rejoined another cluster or the tumor margin. Both were most frequent under intermediate adhesion and high motility, where mechanical interactions allowed partial cohesion and dynamic rearrangement, as illustrated in S4 Video. Cluster trajectories, persistence status, and splitting and merging events across all parameter combinations and replicates can be explored interactively via the (Interactive Cluster Trajectory Explorer) https://sheriffacode.github.io/Cluster_Trajectory_Explorer/ described in S2 Appendix.

These results align with experimental observations that collective dissociation is favored in moderately adhesive environments, where leader cells generate traction while followers maintain partial intercellular bonds. The observed enrichment of leader-dominated, intermediate-sized clusters parallels partial epithelial–mesenchymal transition (pEMT), in which cells retain adhesive contacts while acquiring motility. Notably, the cluster size range observed here (mean 4–8 cells, maximum 30–50 cells) is consistent with intermediate-sized multicellular aggregates documented in experimental invasion assays, where such collectives demonstrate enhanced persistence relative to smaller fragments or individual cells. A comprehensive analysis of cluster persistence times across the full parameter space is provided in S11 Fig, showing that the recorded duration is tightly coupled to the same adhesion-motility balance that governs cluster formation. Together, these findings provide a mechanistic explanation for why intermediate adhesion and strong migration create a regime that fosters cluster detachment, leader-driven persistence, and ultimately, metastatic dissemination.