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Nuclear Thermal Propulsion (NTP) will be an enabling technology for future deep space exploration missions. The high energy density of a nuclear reactor allows for the hydrogen propellant to reach exit velocities much greater than what is possible with traditional chemical propulsion, but that high energy density also presents some issues that must be designed around. Most notably, the decay heat produced after shutdown needs to be removed by venting hydrogen through the core. This thesis attempts to determine the minimum amount of hydrogen propellant that must be spent in this way by computationally modelling a portion of the core to predict exactly what mass flowrate is necessary for keeping important components below critical temperatures. In addition, an investigation into the effectiveness of Auxiliary Heat Removal Systems (AHRSs) at reducing cool-down hydrogen usage is reported on. It is found that significant opportunities for cool-down hydrogen reduction exist, both through computational optimization of the flowrate profile and through the inclusion of an AHRS to remove heat by less wasteful means.