The tritium inventory of direct drive inertial fusion energy (IFE) target filling facilities is examined in the interest of minimizing the tritium inventory. A model is described that has been developed to evaluate the tritium inventory of the target filling process as a function of filling and layering parameters, as well as target design parameters. Previous studies by A. Nobile et al. showed that the temperature and the fill system void fraction have a significant effect on the tritium inventory. The current study uses the model to examine the effect of deuterium-tritium (DT) ice layering time and density of the CH foam in the target on the tritium inventory. The study shows that increasing the foam density and decreasing the DT ice layering time significantly reduce the tritium inventory. Fortunately, one-dimensional target design calculations indicate that the foam density in the direct drive target can be increased to ~200 mg/cm3 without significant degradation of the target yield. Having evaluated and minimized the theoretical tritium inventory, calculations were performed with more realistic batch filling scenarios. The inventories associated with "real" filling scenarios approach the theoretical minimum inventory as the number of batches is increased, resulting in tritium inventories that seem acceptable for future IFE target DT filling facilities.