Elastic Free Energy Drives the Shape of Prevascular Solid Tumors

<div><p>It is well established that the mechanical environment influences cell functions in health and disease. Here, we address how the mechanical environment influences tumor growth, in particular, the shape of solid tumors. In an <i>in vitro</i> tumor model, which isolates mechanical interactions between cancer tumor cells and a hydrogel, we find that tumors grow as ellipsoids, resembling the same, oft-reported observation of <i>in vivo</i> tumors. Specifically, an oblate ellipsoidal tumor shape robustly occurs when the tumors grow in hydrogels that are stiffer than the tumors, but when they grow in more compliant hydrogels they remain closer to spherical in shape. Using large scale, nonlinear elasticity computations we show that the oblate ellipsoidal shape minimizes the elastic free energy of the tumor-hydrogel system. Having eliminated a number of other candidate explanations, we hypothesize that minimization of the elastic free energy is the reason for predominance of the experimentally observed ellipsoidal shape. This result may hold significance for explaining the shape progression of early solid tumors <i>in vivo</i> and is an important step in understanding the processes underlying solid tumor growth.</p></div

3D rendering of an oblate ellipsoidal tumor.

<p>The 3D rendering, created using images taken with a light sheet fluorescence microscope, is rotated about the vertical axis in this sequence of images. Blue: Hoechst-stained nuclei; Red: E-cadherin. Scale bar is 90 µm.</p

Elastic free energy landscapes of the tumor-gel system.

<p>The landscapes are plotted <i>versus</i> the tumor's ellipsoidal axes ratios and . The origin has been shifted, so that the maximum energy is zero in each case: (<b>a</b>) For . The oblate shapes ( =  = 3) are the low energy states. This surface has been generated from 576 separate nonlinear elasticity computations of tumors of aspect ratios and varying between 1 and 3, growing in a gel. (<b>b</b>) The same as (a), but for . Note the flatness of the landscape relative to (<b>a</b>). Spherical shapes are not penalized strongly for , but are strongly penalized for .</p

The stress field created by tumor growth when

<p><b>.</b> The growth volume ratio . All normal stress components, (——), are equal in a spherical tumor of initial radius 50 µm. The maximum compressive stress is −1300 Pa. However, in an ellipsoidal tumor with axes µm, the () component, has a maximum compressive stress of −1200 Pa, and the () component has a maximum compressive stress of −1180 Pa.</p

Oblate ellipsoidal tumors have a similar distribution of orientations regardless of viewing direction.

<p><b>a</b>. The relationship between tumor rotation angle, , and projected aspect ratio, , as an oblate ellipsoid with maximum aspect ratio of 3 (  =  3) is rotated about its () axis. <b>b</b>. The tumor rotation angle was calculated from the projected aspect ratio (part <b>a</b>) for tumors grown in 1% agarose for 30 days. Top image: observation perpendicular to the plane of the cell-culture well, Bottom image: side view perpendicular to a physical cross-section of the gel made with a scalpel blade (out-of-focus cut marks are visible in the gel) Scale bars are 200 µm.</p