Competition between epithelial tissue elasticity and surface tension in cancer morphogenesis

Abstract

We derive a continuum mechanical model to capture the morphological changes occurring at the pretumoral stage of epithelial tissues. The proposed model aims to investigate the competition between the bulk elasticity of the epithelium and the surface tensions of the apical and basal sides. According to this model, when the apico-basal tension imbalance reaches a critical value, a subcritical bifurcation is triggered and the epithelium attains its physiological folded shape. Based on data available in the literature, our model predicts that pretumoral cells are softer than healthy cells.

Introduction

Epithelium is one of the most widespread type of tissue in living things. It may appear with different shape and function, as a mono or multi-layer of cells covering and protecting the inner parts of tissues and organs. These layers are in the shape of flat sheets in the case of skin, or in the shape of corrugated and folded membranes in stomach and intestine, where they give rise to villi and crypts. These corrugations increase the exchanging area, favoring the secretion of enzymes and absorption of nutrients. During embryogenesis epithelial tissues differentiate from all the three embryonic cell layers, undergoing extensive and precise morphological changes, which result in complex folding patterns.

Epithelial morphogenesis is characterized by a highly complex chemo-mechanical phenomenology which is not comprehensively understood yet, and whose review is out of topic of the present paper. Despite this complexity, the purely mechanical aspects of these processes are key in determining tissural architecture. In particular, it is known that epithelial folding can be the manifestation of a mechanical instability triggered by the contractile action of a meshwork of cross-linked actin filaments acting in the proximity of the apical and basal membranes of the cells (see the sketch in Fig. 2 from Section 2.2 of the present paper).

Epithelia are a common site of tumor onset: Carcinomas, arising from the epithelium, represent more than 80% of the cancer-related deaths in the Western world (Weinberg, 2013). In particular, Pancreatic Ductal Adenocarcinoma (PDAC) is the most lethal of the common cancers, without effective therapeutic option except surgery (Therville et al., 2019).

Perhaps the earliest attempt to understand the mechanical basis of epithelial folding dates back to the work of Lewis (1947), who came up with a two-dimensional physical model consisting of a linear framework of pinned bars, representing an epithelium sheet, with elastic cables stretched on the top and bottom side, which would mimic the apical and the basal tensions. More recently, several mechanical models have been devised: from the 2D and 3D vertex models (Honda and Eguchi, 1980, Latorre et al., 2018, Misra et al., 2016, Misra et al., 2017, Nagai and Honda, 2001, Nestor-Bergmann et al., 2018), to continuum models (Bielmeier et al., 2016, Murisic et al., 2015), which picture an epithelial layer as a planar rod or as a shell. Among of the latter, some recent works investigate the epithelial tissue morphogenesis as triggered by buckling instability (Balbi et al., 2020, Balbi et al., 2015, Ben Amar and Jia, 2013, Carotenuto et al., 2021, Drasdo, 2000, Fraldi et al., 2015, Hohlfeld and Mahadevan, 2011, Li et al., 2012, Salbreux and Jülicher, 2017, Shraiman, 2005) or by differential intraepithelial tensions (Hannezo et al., 2017, Krajnc et al., 2013, Papastavrou et al., 2013, Sui et al., 2018). In Krajnc et al., 2013, Krajnc and Ziherl, 2015 and Štorgel, Krajnc, Mrak, Štrus, and Ziherl (2016) apico-basal differential tension is shown to be enough to produce folded configurations in longitudinal epithelial sheets. In Krajnc and Ziherl (2015) a continuum model, derived from area- and perimeter-elasticity (APE) models (Farhadifar, Röper, Aigouy, Eaton, & Jülicher, 2007), is proposed for the healthy epithelium.

Concerning the connection with tumor morphogenesis, as reviewed in Fiore, Krajnc, and Quiroz (2020), the available models address multiple mechanical factors as responsible for the disruption of normal tissue architecture besides the abnormal acto-myosin concentration gradient from basal to apical region (Messal, Alt, & Ferreira, 2019), which is the mechanical effect that we address in the present paper; in particular, other factors include cancer cells proliferation (Bielmeier et al., 2016), lateral cell adhesion (Hannezo et al., 2017), elasticity of the basement membrane (Krajnc & Ziherl, 2015).

We model an epithelial monolayer as a two dimensional thin body equipped with a bulk and a surface energy at the apical and basal sides. These energetic terms are in competition: the former favors the undeformed configuration; the latter induces bending when the apical and basal energies are imbalanced. By dimension reduction, based on a kinematic Ansatz allowing for thickness extension, we arrive at a one-dimensional model of a nonlinear elastic rod whose equilibria are governed by the competition of the aforementioned energetic contributions.

In our model two dimensionless key parameters are introduced, $\gamma$ and $\sigma$: the former is a measure of the relative importance of surface energy compared to bulk energy; the latter is a measure of the imbalance between apical and basal tensions.

As $\gamma$ grows, surface energy becomes more important: the apical and basal sides shorten and, in turn, the thickness increases. A growth of $\sigma$ favors curved configurations.

We formulate a nonlinear equilibrium problem that admits in principle a manifold of solutions. The rectilinear configuration is a solution of this problem for every choice of $\gamma$ and $\sigma$. For $\gamma$ small, bulk elasticity prevails over surface tension and there is no other solution except the rectilinear one; for $\gamma$ large enough, there exists a critical value ${\sigma }_c$ of the parameter $\sigma$ where bifurcation from the rectilinear configuration occurs.

A careful analysis, performed through the Lyapunov–Schmidt decomposition, reveals that the bifurcation is subcritical. This is confirmed by our numerical calculations.

Our model predicts a distinctive mechanical behavior of pre-cancerous cells. Based on data available in Messal et al. (2019), we estimate the pretumoral tissue softening for pancreatic Neoplasia. This result is in accordance with elastographic measures in Therville et al. (2019), and confirms that transformed cells are softer than healthy cells.

In Section 2 we derive our model by prescribing the geometry and the underlying kinematical hypotheses. We specify the form of the bulk and surface energy, and we introduce an incompressibility constraint. We identify the relevant dimensionless parameters, we derive the equilibrium equations, and we formulate a boundary-value problem.

In Section 3 we study the loss of positivity of the elasticity tensor and we determine the critical value of the apico-basal tension imbalance, along with the bifurcation mode.

In Section 4 we perform a detailed bifurcation analysis by means of the Lyapunov–Schmidt decomposition, and we identify the type of bifurcation. We accompany our analysis with a numerical calculation.

In Section 5 we discuss the implication of our model concerning the softening which accompanies incipient tumorigenesis.