Coevolution of solid stress and interstitial fluid pressure in tumors during progression: implications for vascular collapse

Abstract

The stress harbored by the solid phase of tumors is known as solid stress. Solid stress can be either applied externally by the surrounding normal tissue or induced by the tumor itself due to its growth. Fluid pressure is the isotropic stress exerted by the fluid phase. We recently showed that growth-induced solid stress is on the order of 1.3 to 13.0 kPa (10-100 mmHg)--high enough to cause compression of fragile blood vessels, resulting in poor perfusion and hypoxia. However, the evolution of growth-induced stress with tumor progression and its effect on cancer cell proliferation in vivo is not understood. To this end, we developed a mathematical model for tumor growth that takes into account all three types of stresses: growth-induced stress, externally applied stress, and fluid pressure. First, we conducted in vivo experiments and found that growth-induced stress is related to tumor volume through a biexponential relationship. Then, we incorporated this information into our mathematical model and showed that due to the evolution of growth-induced stress, total solid stress levels are higher in the tumor interior and lower in the periphery. Elevated compressive solid stress in the interior of the tumor is sufficient to cause the collapse of blood vessels and results in a lower growth rate of cancer cells compared with the periphery, independently from that caused by the lack of nutrients due to vessel collapse. Furthermore, solid stress in the periphery of the tumor causes blood vessels in the surrounding normal tissue to deform to elliptical shapes. We present histologic sections of human cancers that show such vessel deformations. Finally, we found that fluid pressure increases with tumor growth due to increased vascular permeability and lymphatic impairment, and is governed by the microvascular pressure. Crucially, fluid pressure does not cause vessel compression of tumor vessels.

Figure 1

Tumor opening measurement. After excision of the tumor (left), a partial cut was performed along the longest axis ~80% of the thickness. The cut released the growth-induced stress and the tumor deformed as a result of stress relaxation (right). We then measured the resulting deformation (i.e., tumor opening) between the two halves of the tumor. The figure shows a Mu89 human melanoma.

Figure 2

Tumor opening (left) and estimated residual stress ratio at the periphery and the interior of a tumor (right) as a function of the relative volume (V/V_o) for Mu89, E0771 and AK4.4 tumors. Solid line depicts the least squared fit of Equation (B) to the calculated values of λ_r.

Figure 3

(A) Spatial distribution of the total radial solid stress in the tumor and the surrounding normal tissue for four different time points. Vertical lines on the plot depict the interface between tumor and normal tissue. (B) Radial solid stress at day 18 for three different patterns for the evolution of growth-induced stress: absence of residual stress, low growth-induced stress (E0771) and high growth-induced stress (Mu89). The tumor is taken to be near incompressible with stiffness 20kPa. (C) Spatial distribution of the total circumferential solid stress in the tumor and the surrounding normal tissue for four different time points. (D) Circumferential solid stress at day 18 for two different patterns for the evolution of growth-induced stress: absence of residual stress and high growth-induced stress (Mu89). Only the solid phase is considered for these simulations, the tumor is taken to be near incompressible with stiffness 20kPa.

Figure 4

(A) Spatial distribution of the growth stretch ratio, λ_g, during tumor progression and the effect of growth-induced stress. Residual stretch ratio, λ_r, was taken from the Mu89 data. (B) Relative cancer cell growth between the tumor periphery and interior for three different patterns for the evolution of growth-induced stress: absence of residual stress, low residual stress (E0771) and high growth-induced stress (Mu89).

Figure 5

(A) Profile of radial and circumferential components of the solid stress in tumors. As the tumor grows, solid stresses (indicated by the arrows) become stronger and result in further radial compression of the entire tumor, circumferential compression in the interior, and circumferential tension in the tumor periphery. (B) Schematic of the direct and indirect effects of solid stress on tumor growth and migration. Solid stress (direct) and the resulting hypoxia due to vessel compression (indirect) reduce proliferation rate, increase apoptosis and enhance the invasive and metastatic potential of cancer cells.

Figure 6

Histological sections of a murine tumor (panel A) and different human tumors show the deformation of peritumoral arterioles and venules. Panel A: murine liver tumor, the arrow shows the position of the vessel. Panel B: esophageal carcinoma, panel C: salivary duct carcinoma, panel D: undifferentiated liposarcoma, panel E and F: differentiated pancreatic neuroendocrine tumors. The arrows show the location of the tumor. (scale bar = 100 µm).

Figure 7

(A) Evolution of IFP as a function of time. (B) Spatial distribution of IFP for four different values of the parameter $\theta =\sqrt{\frac{L_{p}S}{KV}}$. θ=75 corresponds to the baseline paremeter values given in the supplement, while θ=10 corresponds to a vascular density of just 4 cm⁻¹ keeping the rest of the parametrs the same.