The growth dynamics of avascular tumors in a microchannel bioreactor is investigated. A three-dimensional flow and nutrient transport model, incorporating the multicellular tumor spheroid (MTS) growth model, has been developed to study the influence of nutrients (oxygen and glucose) supply and distribution on the MTS growth. Numerical simulations based on the EMT6/Ro tumor cells show that the continuous-flow perfusion is more efficient to deliver nutrients to the MTS than the diffusion-only static culture. It is further demonstrated that as long as there is bulk flow, the growth of a single tumor spheroid at the early stage is insensitive to the flow velocity and the channel size. For multiple tumor spheroids in the same microchannel, however, increasing the perfusion velocity can improve the nutrient environment for the disadvantageous downstream tumor spheroid. The flow shear stress exerting on the MTSs in the current microchannel bioreactor is estimated to be far below the critical value to affect the MTS growth, which means that there is still much room for increasing perfusion velocity to satisfy the higher nutrient requirement by the growing tumor spheroids.