Modeling and Simulation of Avascular Tumor Growth using a Level Set Method

A major challenge in matrix-metalloproteinase (MMP) target validation and MMP-inhibitor-drug development for anti-cancer clinical trials is to better understand their complex roles (often competing with each other) in tumor progression. For this purpose, we have developed avascular tumor models capable of examining the complex interaction between avascular solid tumors and their host environment and explored the impact of directed cell motion (chemotaxis and haptotaxis) on the tumor growth and morphology. We have presented three increasingly sophisticated mathematical models of solid tumor growth and their numerical solutions in two spatial dimensions. First, we have developed a continuum model and have examined the effects of haptotaxis due to extracellular matrix (ECM) gradients (caused by degradation of ECM by MMP) on tumor progression and morphology for different cell-cell and cell-ECM adhesion (modeled as surface tension), proliferation and apoptosis rates of tumors growing in nutrient-rich and low-nutrient environments. Morphological instabilities are observed for non-necrotic tumor growth in low-nutrient environment due to the effect of haptotaxis, with the no. of fingers increasing with decreasing surface tension, consistent with the results of previous hybrid (discrete-continuum) models. On the other hand, haptotaxis causes no distinct effect on morphology for fingering growth in nutrient-rich environment. Both chemotaxis (due to nutrient gradients) and haptotaxis lead to shape instability for both low cell proliferation and high apoptosis rates. Therefore inducing high apoptosis during cancer therapy may not be successful for chemotaxis/haptotaxis dominated tumor growth. Next, we have developed a continuum model to explore the growth-inhibiting influences of the gradients of soluble fragments of ECM through chemotaxis on the progression and morphology of a tumor growing in nutrient-rich and nutrient-poor microenvironments with variations of surface tension and for low cell proliferation. A parametric study has been conducted to explore the effects of varying ECM degradation rate, the production rate of matrix degrading enzymes (MDE), and the conversion factor of ECM into soluble ECM. We find that chemotaxis strongly influences tumor growth and morphology, and the instabilities caused by tumor cell proliferation and haptotactic movements can be completely damped out by enforcing the extent of chemotaxis. However, the influence of chemotaxis and the above factors is found to be stronger in low-nutrient environments than that in nutrient-rich environments. As the extent of chemotaxis increases, the effects of cell adhesion on tumor growth and shape becomes negligible. For low cell mitosis, chemotaxis may causes the tumor to shrink, as the extent of chemotaxis increases, but in some cases chemotaxis may initiate shape instability Finally we have developed a continuum model to examine the influence of ECM degradation by membrane type MMP (MT-MMP) through haptotaxis in low and high nutrient environment with different surface-tensions, proliferation and apoptosis rates. We compare these effects with that of MMP-mediated effects and find that, in the absence of necrosis, ECM degradation by MMP gives faster growth and more fingers than that for MT-MMP for low-nutrient environment, even with low proliferation rate, whereas they both lead to similar unstable morphologies for high apoptosis rates. There is no significant differences in the morphologies for MMP and MT-MMP mediated proteolysis for high-nutrient environment. Similarly, we get no significant differences in the morphologies for necrotic tumor growth in both environments although MT-MMP gives faster growth rate. We have solved the models using a level set based interface capturing method and conducted detailed investigation of the effects of MDEs on tumor growth dynamics and morphology over a broad range of biophysical parameters. Our models can predict the tumor growth and morphology due to both promoting and inhibitory effects of MDEs with variations of tissue environment and will, therefore, be helpful to design combination therapies including selective MDE inhibition.