In this study we address a novel design of a planar memristor and investigate its biocompatibility. An experimental prototype of the proposed memristor assembly has been manufactured using a hybrid nanofabrication method, combining sputtering of electrodes, patterning the insulating trenches, and filling them with a memristive substance. To pattern the insulating trenches, we have examined two nanofabrication techniques employing either a focused ion beam or a cantilever tip of an atomic force microscope. Inkjet printing has been used to fill the trenches with the functional titania ink. The experimental prototypes have qualitatively demonstrated memristive current-voltage behavior, as well as high biocompatibility.