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We develop an existence and regularity theory for solutions to a geometric free boundary problem motivated by models of tumor growth. In this setting, the tumor invades an accessible region $D$, its motion is directed along a constant vector $V$, and it cannot penetrate another region $K$ acting as an obstacle to the spread of the tumor. Due to the non variational structure of the problem, we show existence of viscosity solutions via Perron's method. Subsequently, we prove interior regularity for the free boundary near regular points by means of an improvement of flatness argument. We further analyze the boundary regularity and we prove that the free boundary meets the obstacle as a $C^{1,\alpha}$ graph. A key step in the analysis of the boundary regularity involves the study of a thin obstacle problem with oblique boundary conditions, for which we establish $C^{1,\alpha}$ estimates.