Computational modeling of 3D tumor growth and angiogenesis for chemotherapy evaluation

Abstract

Solid tumors develop abnormally at spatial and temporal scales, giving rise to biophysical barriers that impact anti-tumor chemotherapy. This may increase the expenditure and time for conventional drug pharmacokinetic and pharmacodynamic studies. In order to facilitate drug discovery, we propose a mathematical model that couples three-dimensional tumor growth and angiogenesis to simulate tumor progression for chemotherapy evaluation. This application-oriented model incorporates complex dynamical processes including cell- and vascular-mediated interstitial pressure, mass transport, angiogenesis, cell proliferation, and vessel maturation to model tumor progression through multiple stages including tumor initiation, avascular growth, and transition from avascular to vascular growth. Compared to pure mechanistic models, the proposed empirical methods are not only easy to conduct but can provide realistic predictions and calculations. A series of computational simulations were conducted to demonstrate the advantages of the proposed comprehensive model. The computational simulation results suggest that solid tumor geometry is related to the interstitial pressure, such that tumors with high interstitial pressure are more likely to develop dendritic structures than those with low interstitial pressure.

Figure 1. Multi-scale modeling system structure.

Figure 2. An illustration showing one cell located in the center of cube surrounded by 26 neighboring grid points.

The potential directions for spatial transition are calculated according to pressure gradients along these 26 directions.

Figure 3. Branching Hotpoint (BH)-induced tumor vasculature branching.

Green dots indicate area of enhanced concentration of stimuli for vessel branches.

Figure 4. Tumor cell number over time. Blue: total cells; Red: active tumor cells; Green: quiescent tumor cells; Black: necrotic tumor cells.

Figure 5. Tumor volume and morphology changes during progression including exponential growth, linear expansion, stasis, and secondary growth processes (T1–T4).

Brown region: Viable cells; Black region: Necrotic cells.

Figure 6. Angiogenic sprouting and vessel maturation during tumor growth.

New vessel branches and sprouts occur at vessel branching hotpoints (VBHs) in response to changes in TAF concentration and interstitial pressure. These vessels grow towards hypoxic regions in order to provide independent blood supply. Vessel diameter as well as vessel density increases with time, shown here at timepoints. Note that the vessels are absent from tumor center, due the presence of a necrotic cell core.

Figure 7. Influence of metabolite and growth factor distribution on tumor properties.

(a) Vascularized 3D solid tumor morphology on Day 45; (b) Predicted oxygen distribution; (c) Predicted carbon dioxide distribution; (d) Predicted TAF distribution; (e) Cell activity distribution; (f) CVE distribution; (g) Interstitial pressure distribution; MDE distribution is similar to tumor pressure field. (Intensity: Red (high) → Blue (low)).

Figure 8. Predicted tumor morphology as a function of interstitial pressure.

Low pressure tumors (40 mmHg) develop a rounded morphology (left) whereas high pressure tumors (60 mmHg) develop a dendritic morphology. Simulated time period: Day = 45.

Figure 9. Solid tumor growth simulation with angiogenesis from a virtual 3D vasculature.

The cross-sectional plane shows the nutrient availability within the simulation area.

Figure 10. Tumor morphology and drug distribution following three different drug administration concentrations (0.1, 1, and 10 mol/m³), shown at normalized scale.

In our model, drug is inserted at day 40. Two figures in the top left figures show tumor growth without drug. The rest figures show the effect of drug on tumor on the same day (55) with different drug concentrations (shown on the right figures with color bar). Brown region denotes living cells and black region indicates necrotic cells.