Does loss of cell polarity alone drive the geometry of epithelial carcinogenesis in three dimensions? Mechanical extension, quantitative phase behavior, and empirical calibration of the Fleury-Schwartz lattice model

Fleury and Schwartz (Fractals 11, 397-414, 2003) showed with two-dimensional lattice simulations that a single cellular defect - a clonally inherited probability $P$ of misoriented (out-of-plane) mitosis, with growth rate, adhesion and all other properties held normal - reproduces the qualitative morphological sequence of squamous carcinogenesis (apparently normal thickened epithelium, metaplasia, dysplasia, carcinoma in situ, invasive fractal-dendritic carcinoma) and generates clonal competition in which severe clones suppress weaker ones. The unresolved questions arising from that two-dimensional demonstration are: (i) whether this faceted-to-rough-like transition survives in a fully three-dimensional model that includes tissue elasticity and surface tension, and at what capillary smoothing scales; (ii) whether the transition has sharp control-parameter thresholds, a phase diagram, and critical scaling (finite-size scaling, exponents), rather than being a gradual crossover; (iii) the biological origin and measurable ranges of the misorientation probability $P$ and the epithelial-rearrangement probability $k$, with statistical validation against histological datasets; (iv) replacement of the abstract growth-promoter field by explicit reaction-diffusion or mechano-chemical kinetics, and robustness of the morphological sequence to added biological complexity (extracellular matrix, vasculature, non-epithelial cell types, cell-size/shape heterogeneity); and (v) dynamical and clinical outcomes (dormancy, spontaneous regression, effects of resecting dominant clones) that require cell death, immune response, and stochastic reversal of polarity defects. No published work constructs such a three-dimensional polarity-loss model, quantifies or calibrates the transition, or refutes it; the problem remains open.

Background

Most cancers of the skin and of the lining of organs arise in epithelia, the sheets of tightly packed cells that cover surfaces. A stratified squamous epithelium, such as that of the skin or the esophagus, is organized in layers on a basement membrane, and its cells are polarized: apicobasal polarity is the asymmetric organization of a cell into an outer (apical) and an inner (basal) face. The plane of cell division - the orientation of the mitotic spindle relative to the tissue layer - is a polarized, actively controlled property; in stratified epithelia, dividing basal cells normally orient so that daughter cells remain within the layer. Squamous carcinogenesis follows a stereotyped morphological sequence: metaplasia (one differentiated cell type replaced by another, often with thickening), dysplasia (disordered maturation and layering), carcinoma in situ (fully disordered cells still confined by the basement membrane), and invasive carcinoma (cells breaching the membrane into surrounding tissue, typically along irregular, often fractal-shaped fronts).

Before the source paper, physical models of tumor shapes used stochastic growth processes from statistical physics: the Eden model (compact growth by random addition at the surface), invasion percolation (growth through a heterogeneous medium), and diffusion-limited aggregation (dendritic growth driven by a field obeying the Laplace equation, written inline as $\nabla^2\Phi=0$, where $\Phi$ is the concentration of a diffusible promoter). These reproduce rounded or fractal tumor outlines but contain no representation of cell polarity or of the epithelial division plane.

Fleury and Schwartz (2003) introduced two two-dimensional lattice models in which the only abnormality of malignant cells is division-plane misorientation. In Model 1, an epithelium grows on a rectangular lattice; a diffusible growth promoter is sampled by random walkers (whose continuum limit is a Laplacian field); normal cells divide only in the plane of the sheet, subject to contact inhibition (division blocked when lateral neighbors are occupied) and mitotic rescue (normal cells fill kinks left by misoriented neighbors); malignant cells inherit a probability $P$ of out-of-plane (misoriented) division. Model 2 adds a fast stochastic epithelial rearrangement (motility) event with probability $k$, motivated by the fact that epithelial cells can rearrange much faster than they divide. Sweeping $P$ from $5\times10^{-4}$ to $5\times10^{-2}$ generated the full morphological sequence - apparently normal but already diseased thickened tissue, then metaplastic, dysplastic, and carcinoma-in-situ configurations, then invasive fractal-dendritic patterns - which were matched qualitatively to histological micrographs. The change of morphology with $P$ was likened to the faceted-to-rough transition of growing crystals: below its roughening transition a crystal grows a flat facet, while above it the surface becomes rough and dendritic. The capillary length is the length scale below which surface tension (the contractility of an interface) smooths shape fluctuations. Two clonal-competition mechanisms appeared: active smothering, in which a more aggressive clone recruits trains of normal epithelial cells ahead of itself and occupies territory the weaker clone would otherwise colonize (with a heuristic estimate that the blocking distance scales as $1/P$), and passive diffusive screening of the growth-promoter field. The model suggested that severe clones kill or suppress weaker ones, and that low-$P$ tissue can be long-lived, macroscopically normal, yet malignant-cell-bearing (a dormant, possibly regressive state) - although the model itself implemented no cell death and no reversal of the malignant state.

The demonstration was, however, two-dimensional and mechanically ungrounded: no tissue elasticity, no surface tension or capillary-length effects, no extracellular matrix, blood vessels, or non-epithelial cells; each morphology class came from a single simulation run without statistics; no order parameters (quantitative measures whose values distinguish the morphological regimes), thresholds, or critical exponents were measured; the parameters $P$ and $k$ were not tied to measurable biology; the growth promoter remained an abstract field rather than a specified biochemical or mechanical signal; and the dynamical and clinical implications (regression probabilities, effect of resecting the dominant clone) were unmodeled.

In parallel, in vivo biology has established that polarity loss causally promotes malignancy: in Drosophila, mutation of apicobasal polarity genes combined with oncogenic Ras converts benign growths into invasive, metastatic tumors via JNK (c-Jun N-terminal kinase, a stress-signaling pathway) signaling and downregulation of E-cadherin (the cell-cell adhesion molecule that holds epithelial cells together) (Igaki, Pagliarini and Xu 2006); Lee and Vasioukhin (2008) reviewed cell-polarity genes as non-canonical tumor suppressors; and Jeyanathan et al. (2025) proposed a feedforward mechanism linking cell division itself to transient mitotic polarity loss as a driver of epithelial tumor progression. On the modeling side, three-dimensional, mechanics-aware cell-based models of epithelial populations and tumors now exist (Galle, Loeffler and Drasdo 2005 on deregulated proliferation and apoptosis; Drasdo et al. 2010 on steps from benign tumor to invasive cancer), and Azote et al. (2026) built a three-dimensional vertex model of stratified epithelia - a representation in which cell-cell interfaces are the faces of a deformable mesh - coupling division orientation to interfacial tensions and tissue stiffness in development. The kinetic roughening of tumor growth fronts has been quantified in a different framework (Bru et al. 1998). None of these addresses the specific question raised by the Fleury-Schwartz model.

Problem Statement

Determine whether loss of cell polarity alone - implemented strictly as a clonally inherited increase in the probability $P$ of misoriented (out-of-plane) mitosis, with growth rate, adhesion, and all other cell and tissue properties held at their normal values - is sufficient to generate the morphological sequence of epithelial carcinogenesis (apparently normal thickened epithelium, metaplasia, dysplasia, carcinoma in situ, invasive carcinoma) and the associated clonal-competition dynamics, once the model is made biophysically realistic. The unresolved questions, kept in their full generality, are:

(1) Dimensionality and mechanics. Construct a fully three-dimensional, cell-resolved model of a stratified epithelium that includes tissue elasticity and surface tension with their associated capillary smoothing scales, in which division-plane misorientation is the only abnormality, and determine whether the faceted-to-rough-like transition from ordered, stratified, contact-inhibited tissue to disordered, invasive, fractal-dendritic morphology persists, and on what length scales surface tension smooths the fronts.

(2) Quantitative phase behavior. Establish whether this transition is quantitatively sharp: compute the phase diagram in the space of control parameters (misorientation probability $P$, epithelial rearrangement probability $k$, mechanical parameters), locate the boundaries between the morphological regimes, and perform finite-size scaling (the standard statistical-physics method of extracting critical behavior and exponents from the dependence of observables on system size) to determine whether the transition exhibits critical behavior and universal exponents, or is instead a gradual crossover.

(3) Parameter identification and empirical validation. Identify the biological correlates and measurable ranges of $P$ (e.g. distributions of mitotic spindle orientation in normal, dysplastic, and malignant epithelium) and $k$ (epithelial rearrangement or restitution kinetics), and statistically compare model-generated morphologies against histological datasets, so that the qualitative correspondence becomes predictive validation (including whether the morphological classes can be distinguished from those generated by alternative mechanisms).

(4) Morphogen identity and added biological complexity. Replace the abstract growth-promoter field (random walkers or Laplacian field) with explicit reaction-diffusion or mechano-chemical kinetics for a specified biochemical or mechanical promoter, and test the robustness of the morphological sequence and of the clonal-competition mechanisms when extracellular matrix, blood vessels, non-epithelial cell types, and realistic cell-size and shape heterogeneity are added.

(5) Dynamical and clinical outcomes. Extend the model with cell death, immune response, and stochastic reversal of polarity defects to examine the long-lived dormant state, the probability of spontaneous regression, the consequences of resecting a dominant clone for resurgence of suppressed clones, and the relation between modeled regression probabilities and observed clinical regression rates.

These facets share one common scientific objective: to establish, in a realistic and measurable setting, whether the single-mechanism geometric hypothesis of the source - that polarity loss alone drives the geometry of epithelial carcinogenesis - is correct, and to make it quantitative and falsifiable. Facets (1)-(2) concern the existence, sharpness, and scaling of the geometric transition; facets (3)-(4) ground the model in measurable biology; facet (5) derives the model's dynamical and clinical consequences from the same core. Facet (5) is a model-dependent extension (it requires adding cell death, immune response, and stochastic reversal of polarity defects). Facets (3)-(5) are extensions that ground the model in measurable biology and derive its dynamical and clinical consequences from the same core.

References

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