Fig 1.
The top left part represents a relatively small tumor which is undergoing the vascular switch. The tumor is radially divided into three parts, which represent the proliferative rim (outer ring), the hypoxic zone (mid part), and the necrotic core (inner area). A small area of the tumor is surrounded by a gray solid line whose interior is zoomed in. In the zoomed-in area, we can observe tumoral cells colored with different brown tones, according to their condition of necrotic, hypoxic, or proliferative cells. The red area shows the capillaries, which are lined by endothelial cells as shown in the plot. The new sprouts are led by tip endothelial cells (TECs), which are endowed with filopodia to better explore their surroundings [10, 15]. The picture also shows how the capillaries release nutrients that diffuse throughout the tissue. Similarly, hypoxic cells release vascular endothelial growth factor (VEGF), which eventually binds to surface receptors located in the membrane of endothelial cells (VEGFR). Finally, the figure shows how TECs overexpress the protein Delta-like ligand 4 (Dll4). This protein binds to the Notch receptors of nearby endothelial cells, preventing them from also becoming TECs [12, 13].
Fig 2.
Chemical free energy of the tumor.
The tumor chemical free energy, , is defined as a double-well potential formed by the sum of a symmetric contribution (g) and a non-symmetric function (h). The latter is multiplied by a tilting function m, that depends on the nutrient concentration. On the right hand side of the plot, σn−h and σh−v represent values of the nutrient concentration that define, respectively, the necrotic-hypoxic and hypoxic-viable thresholds.
Table 1.
In silico values of the parameters used in the proposed model.
Fig 3.
Secretion rate of tumor angiogenic factor ().
Tumor angiogenic factor is released by hypoxic tumor cells, that is, those whose nutrient availability is not enough for proliferation, but higher than the apoptotic threshold (σn-h > σ > σh-v).
Fig 4.
The domain of the problem represents a rectangular tissue of size 2625 µm × 2025 µm with a circular initial tumor (brown) and two capillaries (red). Using the symmetries of this setup, the simulations may be performed on a quarter of the tissue, the domain Ωq.
Fig 5.
Simulation of avascular tumor growth.
The amount of nutrient (blue scale) released from the capillary at the bottom edge is not enough for tumor cells to proliferate. Thus, the tumor remains at a constant size within usual experimental time scales (tumor area graph) and shrinks at very long time scales (brown contour lines). By day 1054 (solid brown), the tumor radius has been reduced to approximately 67% of its initial value.
Fig 6.
Hypoxic tumor cells (dark brown; see color scales in the bottom left corner) release tumor angiogenic factor (not shown in the plot for clarity) that promotes the growth of capillaries (red). The new vasculature brings additional nutrients (blue scale) to the tumor favoring its growth (see the tumor area evolution in the top right inset). The top left inset shows through contour lines the time evolution of the tumor and the capillaries that have penetrated it. The bottom right inset shows a tip endothelial cell that is about to anastomose with other capillary and the nutrient distribution in the surroundings of the neovasculature. As expected, the nutrient concentration decreases with the distance to the capillaries.
Fig 7.
Simulation of tumor-growth reduction by blocking Dll4.
The simulation in the top row shows snapshots of the tumor (brown scale), capillaries (red), and nutrient concentration (blue scale) for a typical value of the effective Dll4 distance, namely δ4 = 80. Note that on the right hand side panels we removed the capillaries to allow for a clearer observation of the nutrient distribution. The mid row shows the results of an analogous simulation in which the effective Dll4 distance has been reduced to δ4 = 55 to simulate the negative regulation of the Dll4 signaling pathway. As expected, the simulation shows a denser vascular network. The bottom left panel quantitatively illustrates this point. The denser vascular network, however, does not lead to faster tumor growth, but the opposite (see the bottom central panel). This phenomenon has been observed experimentally and is a consequence of the reduced transport functionality of the vascular network. In our model, the transport capacity is measured by the quantity [see Eq (7)], whose time evolution is shown in the bottom right panel.