Fig. 1. Diagram of the regions in and near a growing tumor: the tumor Ω is comprised of viable (proliferating and quiescent) cells in Ω_V and necrotic cells in Ω_N. The non-cancerous tissue surrounding the tumor, denoted by Ω_H, is affected by the growing tumor, and portions of Ω_H may be encapsulated by the growing tumor. Lastly, the non-cancerous tissue Ω_O is not affected by the growing tumor.

Fig. 2. Representing an ellipse Σ as the zero contour of a level set function .

Fig. 3. Numerical convergence: Tumor morphology at t=0.15 at low resolution (upper left), medium resolution (upper right), and high resolution (lower left). The dark region denotes the necrotic core Ω_N where σN=0.35, and the gray regions show the viable region Ω_V. In the lower right plot, we compare the position of the tumor boundary at low resolution (dotted curve), medium resolution (dashed curve), and high resolution (solid curve).

Fig. 4. Tumor morphological response to the microenvironment. The external tissue nutrient diffusivity D increases from left to right, and the external tissue mobility μ increases from bottom to top. Three major morphologies are observed: fragmenting growth (left), invasive fingering (lower right), and compact/hollow (upper right). All tumors are plotted to the same scale, where the indicated length is .

Fig. 5. Long-time simulation of fragmenting growth into nutrient-poor (D=1), high-mobility (μ=∞) tissue. Plots are in T=10.0 increments, , and A=0.

Fig. 6. Contours of the nutrient concentration near a tumor fragment from the previous figure at T=20.0. The black regions denote nutrient-starved regions where σ<N. Notice the large variation in nutrient level from the left side of the fragment to the right.

Fig. 7. Top row: Evolution of the shape parameter S (solid curves) and length scale LS (dashed curves) for fragmenting tumor growth into nutrient-poor tissue. The left plot is for growth into high-mobility tissue (D=1,μ=∞) and the right plot is for growth into low-mobility tissue (D=1,μ=1). Bottom row: Evolution of the viable and necrotic volume fractions for the high-mobility case (μ=∞; left plot) and low-mobility case (μ=1; right plot).

Fig. 8. Comparison of fragmenting tumor growth into high-mobility tissue (μ=∞; left plots) and low-mobility tissue (μ=1; right plots) at T=60.0 and 70.0.

Fig. 9. Parameter study in G and G_N for fragmenting tumor growth into nutrient-poor, low-mobility tissue (D=1,μ=1). The tumor aggressiveness parameter G increases from bottom to top, and the necrotic degradation parameter G_N increases from left to right.

Fig. 10. The effect of N on fragmenting tumor growth into nutrient-poor, low-mobility tissue (D=1,μ=1): From left to right: N=0.175,0.350, and 0.700. The top row gives the morphology at T=20.0 and the bottom row plots the shape parameter S (solid curves) and length scale LS (dashed curves). G=20,G_N=1, and A=0 for all three simulations.

Fig. 11. Long-time simulation of invasive, fingering growth into nutrient-rich (D=50), low-mobility (μ=1) tissue. Plots are in T=10.0 increments, , and A=0.

Fig. 12. Contours of the nutrient concentration (left) and pressure (right) between growing fingers from the previous simulation at time T=20.0. In the nutrient figure, the black region denotes where σ<N, and in the pressure figure, the boundary of the necrotic core is given by the white curve. Notice that the pressure gradient is primarily parallel to the fingers.

Fig. 13. Evolution of the shape parameter S (top left), length scale LS (top right), perimeter (bottom left), and viable tumor area (bottom right) for invasive, fingering growth into nutrient-rich, low-mobility tissue (thin curves: μ=0.25, thick curves: μ=1). In all plots, dotted lines are for D=50, dashed lines for D=100, and solid lines for D=∞.

Fig. 14. Parameter study in G and G_N for invasive, fingering tumor growth into nutrient-rich, low-mobility tissue (D=50,μ=1). The tumor aggressiveness parameter G increases from bottom to top, and the necrotic degradation parameter G_N increases from left to right.

Fig. 15. Long-time simulation of compact tumor growth into nutrient-rich, high-mobility tissue (D=100,μ=50). Plots are in T=10.0 increments, with G=20,G_N=1, N=0.35, and A=0.0.

Fig. 16. Evolution of the shape parameter S (left) and length scale LS (right) for compact growth into nutrient-rich, high-mobility tissue (thin curves: μ=50, thick curves: μ=∞). In all plots, dotted curves are for D=50, dashed curves designate D=100, and solid curves give D=∞.

Fig. 17. Parameter study in G and G_N for compact tumor growth into nutrient-rich, high-mobility tissue (D=50,μ=∞). The tumor aggressiveness parameter G increases from bottom to top, and the necrotic degradation parameter G_N increases from left to right.

Fig. 18. In vitro experimental evidence from the study by Frieboes et al. (2006b) of predicted tumor morphologies. Lower left: Low glucose, 1% FBS. Lower right: High glucose, 1% FBS. Upper left: Low glucose, 10% FBS. Upper right: High glucose, 10% FBS.

Fig. 19. Discretizing α²u across the interface at x_Σ by a ghost fluid extension to . Care must be taken to enforce the jump boundary conditions [u]=g and [αu·n]=0.