Positron emission tomography (PET) scanners use scintillation crystals to stop high-energy photons. The ensuing lower-energy photons are then detected via photomultipliers. We study the performance of a stack of monolithic siliconpixel detectors as an alternative to the combination of crystals and photomultipliers. The resulting design allows for pitches as small as 100 mu m and greatly mitigates depth-of-interaction problems. We develop a theory to optimize the sensitivity of these and other scanners under design constraints. The insight is complemented by Monte Carlo simulations and reconstructions thereof. Experiments and theory alike suggest that our approach has the potential to move PET closer to the microscopic scale. The volumetric resolution is an order of magnitude better than that of the state of the art and the parallax error is very small. A small-animal scanner is now under construction.