Mechanics of poking a cyst

Abstract

Indentation tests are classical tools to determine material properties. For biological samples such as cysts of cells, however, the observed force-displacement relation cannot be expected to follow predictions for simple materials. Here, by solving the Pogorelov problem of a point force indenting an elastic shell for a purely nonlinear material, we discover that complex material behaviour can even give rise to new scaling exponents in this force-displacement relation. In finite-element simulations, we show that these exponents are surprisingly robust, persisting even for thick shells indented with a sphere. By scaling arguments, we generalise our results to pressurised and pre-stressed shells, uncovering additional new scaling exponents. We find these predicted scaling exponents in the force-displacement relation observed in cyst indentation experiments. Our results thus form the basis for inferring the mechanisms that set the mechanical properties of these biological materials.

Figures (4)

• Figure 1: Mechanics of poking a cyst. (a) A cyst consists of a spherical monolayer of cells surrounding a fluid-filled lumen. Indentation of the cyst by a microbead attached to an AFM cantilever Shen2017 yields the relation between the poking force $F$ and the indentation $e$ of the cyst. (b) The Pogorelov problem pogorelov: A thin hemispherical shell of radius $R$ and thickness $h\ll R$ is indented by a point force $F$. (c) Scaling exponents in the force displacement relation for the unpressurised Pogorelov problem ($p=0$) pogorelovlandaulifshitzaudoly and the pressurised Pogorelov problem (${p\neq 0}$) Vella2012Vella2012PRL. Insets: alternative scaling exponents predicted by the Hertzian contact model Hertzlandaulifshitzaudoly and the models of Fery et al.Fery2004 and Lacorre et al.Lacorre2024.
• Figure 2: Nonlinear Pogorelov problem. (a) Geometry: an indented spherical elastic shell of radius $R$ and thickness $h$ forms a dimple of depth $e$ and radius $\rho$ for $e\gg h$. The dimple ridge, over which the tangent angle changes from $-\alpha$ to $\alpha$, has extent $\delta$. (b) Scaling exponents in the relation between indentation force $F$ and depth $e$: classical results for a linearly elastic shell ($G_2=0$) and results for a purely nonlinearly elastic shell ($G_1=0$). (c) Possible diagrams of scaling exponents in the force-displacement relationship, depending on the linearity $g=G_1/G_2$ and the non-dimensional thickness $\eta=h/R$. The critical non-dimensional indentations $\varepsilon=e/R$ for transitions between scaling exponents are indicated on the axes. (d) Force-displacement relation from finite-element simulations of the Pogorelov problem for a thin shell with $\eta=0.005$, for $g\in [10^{-10},10]$. Dotted lines: analytical approximations of the prefactor of the scaling relation for $e\gg h$, for $G_2=0$ (black) and $G_1=0$ (red). Parameter values satify $C_{10}=C_{01}$, $C_{20}=C_{11}=C_{02}$. (e) Corresponding force-displacement relation for a thick shell ($\eta=0.2$) indented by a point force. (f) Corresponding force-displacement relation for a thick shell indented by a sphere of radius $0.4R$.
• Figure 3: Diagrams of scaling exponents for a pressurized Pogorelov dimple with pre-strain $E_0$, shown for the strongly pressurised, weakly nonlinear case $\mathit{\Pi} \gg g \eta^2$, $g\gg E_0^2$, and for indentations $\varepsilon\ll\eta$. There are four different possible diagrams; the additional conditions in which each diagram occurs are shown above the figure panels, and the scalings of the critical indentations at which the exponents transition are indicated on the axes.
• Figure 4: Observation of scaling exponents in cyst indentation experiments [Fig. \ref{['fig1']}]. (a) Example of force-displacement relation from indentation of a cyst of MDCK cells Shen2017. Experimental data provided by Pingbo Huang and Yusheng Shen. Inset: same data plotted on logarithmic axes, showing a scaling exponent increasing from $1/2$ to $2$. (b) Plot of the local scaling exponent $\beta=\mathrm{d}(\log{F})/\mathrm{d}(\log{e})$ against $e$, estimated by smoothing the raw data in panel (a).