Preparation of tissue engineered (TE) 3D constructs to repair large bone defects is limited by the difficult supply of nutrients and oxygen to cells in the innermost regions of constructs cultured in bioreactors. Poor oxygenation negatively affects cell viability and function. Bioreactor design optimization may help relieve these limitations. Bioreactors in which cells are cultured outside bundles of hollow fiber membranes (HFMBs) are structurally similar to natural bone. HFMB operation in pure diffusion has been reported to suffice for fibroblasts, but is deemed insufficient for bone cells. In this paper, the effect of perfusion flows in the cell compartment on solute transfer was investigated in HFMBs differing in design and operating conditions. HFMBs were designed and operated using values of non-dimensional groups that ensured solutes transfer towards the cell compartment mainly by diffusion; in the presence of low to high Starling flows; in the presence of pulsatile radial flows obtained by periodically stopping the solution flow leaving the bioreactor using a pinch valve. Distribution of matter in cell-free HFMBs was evaluated with tracer experiments in an optimized apparatus. Effectiveness of solute transfer to cell compartment was assessed based on the bioreactor response in terms of the shell volume actively involved in mass transfer (V(MTA)) according to transport models developed specifically for the purpose. V(MTA) increased with increasing Starling flows. In the pulsatile radial flow mode, tracer concentration in the shell increased 3 times faster than at high Starling flows. This suggests that controlled perfusion flows in HFMBs might enable the engineering of large TE bone constructs.