We develop a coupled mathematical model of avascular tumor growth based on porous media mechanics. This comprises of the migration of tumor cells (TCs), the degradation of extracellular matrix (ECM), the transport of matrix-degrading enzymes (MDEs), the seepage of tissue fluid and the supplement and consumption of oxygen. The simulation that a solid tumor grows in the micro-environment composed of the pre-existing capillaries and the surrounding tissues, and the specific property of varying porosity with the growth of TCs in a tumor micro-environment are taken into account. We propose functional coefficients for fluid seepage and oxygen diffusion, and incorporate the convection-diffusion of oxygen and the convection of MDEs. From this modified model the main findings include: first, a solid tumor originating in the inlet region undergoes necrosis in the outlet region because of a low supply of oxygen, while a solid tumor originating in the outlet region undergoes necrosis at the primary site because of overconsumption of oxygen; second, tumors further from capillaries grow faster than tumors close to adjacent capillaries; third, the pre-existing capillaries cause some impact to the transport of those chemical factors involved in tumor growth, and further affect tumor migration and necrosis.
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