Solid Tumors Are Poroelastic Solids with a Chemo-mechanical Feedback on Growth

Abstract

The experimental evidence that a feedback exists between growth and stress in tumors poses challenging questions. First, the rheological properties (the “constitutive equations”) of aggregates of malignant cells are still a matter of debate. Secondly, the feedback law (the “growth law”) that relates stress and mitotic–apoptotic rate is far to be identified. We address these questions on the basis of a theoretical analysis of in vitro and in vivo experiments that involve the growth of tumor spheroids. We show that solid tumors exhibit several mechanical features of a poroelastic material, where the cellular component behaves like an elastic solid. When the solid component of the spheroid is loaded at the boundary, the cellular aggregate grows up to an asymptotic volume that depends on the exerted compression. Residual stress shows up when solid tumors are radially cut, highlighting a peculiar tensional pattern. By a novel numerical approach we correlate the measured opening angle and the underlying residual stress in a sphere. The features of the mechanobiological system can be explained in terms of a feedback of mechanics on the cell proliferation rate as modulated by the availability of nutrient, that is radially damped by the balance between diffusion and consumption. The volumetric growth profiles and the pattern of residual stress can be theoretically reproduced assuming a dependence of the target stress on the concentration of nutrient which is specific of the malignant tissue.

Appendix: Stability of the Homogeneous Solution

The integration of the “stress-modulated growth” illustrated in the previous section predicts a spatially homogeneous solution, parametrically depending on the time-dependent growth rate. The growth $g(R;t)$ is independent of the radial position because the stress is the same everywhere. In such a purely mechanical setting we now study the stability of the homogeneous solution (20) and (25). In other words, the question is whether the spatial inhomogeneity observed in grown spheroids could be produced by the mechanobiological feedback, thus amplifying the spatial perturbations of the stress to yield inhomogeneous growth.

To investigate this hypothesis we consider the following perturbation of the homogeneous solution:

$$\begin{aligned} r(R,t) &= \gamma g_{0}(t) R + \rho (R,t), \quad \gamma g_{0}(t) R \gg \rho (R,t), \end{aligned}$$

(32)

$$\begin{aligned} g(R,t) &= g_{0}(t) + \delta (R,t), \quad g_{0}(t) \gg \delta (R,t), \end{aligned}$$

(33)

where $\gamma$ and $g_{0}(t)$ are solutions of Eqs. (21) and (24), respectively, and $g_{0}(0) = 1$.

When the perturbed solutions are plugged in Eqs. (21) and (24) and only first order terms are retained, the following linear equations are found

$$\begin{aligned} \biggl( \rho ' + 2 \frac{\rho }{R} \biggr) ' &= \gamma \frac{3 - \gamma^{2}}{\gamma^{2} + 1} \delta ', \end{aligned}$$

(34)

$$\begin{aligned} \dot{\delta} &= \biggl( 1 + 2 \frac{g_{0}}{\kappa } -2 \frac{g_{0}}{ \alpha } -4 \frac{g_{0}}{\kappa \gamma^{2}} \biggr) \frac{\delta }{ \tau } + \frac{2}{3} \frac{g_{0}}{\kappa \tau \gamma^{3}} \biggl( \rho ' + 2 \frac{\rho }{R} \biggr) . \end{aligned}$$

(35)

Derivation of the former equation in space, derivation of the latter in time and cross substitution yields

$$\begin{aligned} \dot{\delta }' = \biggl( 1 + 2 \frac{g_{0}}{\kappa } -2 \frac{g_{0}}{ \alpha } - 4 \frac{g_{0}}{\kappa \gamma^{2}} + \frac{2}{3} \frac{g _{0}}{\kappa \gamma^{2}} \frac{3 - \gamma^{2}}{\gamma^{2}+1} \biggr) \frac{ \delta '}{\tau }, \end{aligned}$$

(36)

which determines the evolution in time of the spatial perturbation in the growth $g(t)$. Instability shows up if

$$\begin{aligned} \alpha \kappa \gamma^{2} \bigl(\gamma^{2}+1 \bigr) > 2 g_{0} \biggl( \alpha \biggl( 1 + \frac{4}{3} \gamma^{2} - \gamma^{4} \biggr) + \kappa \gamma^{2} \bigl( \gamma^{2} + 1 \bigr) \biggr) , \end{aligned}$$

(37)

for some $1< g_{0}(t)< g_{e}$, where $0 <\gamma (p_{D})<1$.

The result (37) is negative versus our conjecture: it predicts a stabilization of the system for large enough growth $g_{0}$ which is not in agreement with experiments. If the purely mechanical system is stable, the reported inhomogeneity (large proliferation near the boundary, smaller internally) should instead be explained accounting for the role of nutrients.