Mathematical Modeling of Two-dimensional Unsteady Flow in Growing Tumor

We investigate the problem of unsteady fluid flow in growing solid tumors. We develop a mathematical model for a growing tumor whose boundary is taken as a sphere, and the unsteady fluid flow within the tumor is assumed to be two dimensional with respect to the radial distance and the latitudinal angle in spherical coordinates. The expressions for the time, radial and latitudinal variations of the flow velocity, pressure, and the two investigated drug concentrations within the tumor were determined analytically. We calculated these quantities in the tumor as well as in a corresponding normal tissue. We find, in particular, that blood pressure in the tumor would be higher than that in the normal tissue, and there could be blood flow circulation in the tumor. For a given spatial location in the tumor, the amount of drug delivered to the growing tumor decreases first with time, but then the rate of decrease reduces with further increase in time. The Therapeutic Index, which is a measure of the efficiency of drug delivery in the tumor in the biomedical science, is determined for different values of the parameters and discussed in the absence or presence of the drugs' interactions which may exist in the presence of the two drugs in the tumor. The main results of our model agree with the available experiments.