Buckling Instability in Growing Tumor Spheroids

A growing tumor is subjected to intrinsic physical forces, arising from the cellular turnover in a spatially constrained environment. This work demonstrates that such residual solid stresses can provoke a buckling instability in heterogeneous tumor spheroids. The growth rate ratio between the outer shell of proliferative cells and the inner necrotic core is the control parameter of this instability. The buckled morphology is found to depend both on the elastic and the geometric properties of the tumor components, suggesting a key role of residual stresses for promoting tumor invasiveness.

Figure 1

Morphological evolution of a multicellular tumor spheroid of HeLa cells, developing an undulated contour and a necrotic core (lighter cells). The bar is $100\mu \mathrm{m}$ (top, from Ref. 30). Magnetic resonance sections of a glioblastoma multiforme, an irregular brain tumor characterized by the presence of inner necrosis (bottom).

Figure 2

Marginal stability curves for $g_s/g_c$ (top, thick line) and $m$ (bottom) versus the aspect ratio $R_o/R_i$, calculated for ${\mu }^s/{\mu }^c=1$. The thin solid lines represent the critical thresholds $g_s/g_c$ for different values of the wave number $m=2,\dots ,20$. The inset shows the marginal stability curve for $g_s/g_c$ versus a larger range of $R_o/R_i$.

Figure 3

Logarithmic plot of the marginal stability curves for $g_s/g_c$ versus the shear moduli ratio ${\mu }^s/{\mu }^c$. The solid lines represent the instability thresholds for $R_o/R_i=1.05$, 1.5, 2, 3.

Figure 4

Tumor morphology at the critical thresholds $g_s/g_c=1.33$, $m=20$ for ${\mu }^s/{\mu }^c=1$, $R_o/R_i=1.04$ (left). The numerical solutions (solid lines) and the WKB approximations (dashed lines) for the displacement fields $U$ and $V$ are shown setting $g_c{R}_i=1$, and $ϵ=0.15$ (right).