A review of the existing methods to compute the minimal distance between two ellipsoids has been conducted in order to retain the most adequate one within the context of Particle-Resolved Direct Numerical Simulations for particle-laden flows. First, all methods have been implemented and the corresponding algorithms are reported. Furthermore, a procedure has been systematically suggested to control the error associated with each method, when such control was not explicitly available. In a second phase, a two-ellipsoid configuration where an analytical solution is known has been used to perform an error study. This allows to assess the accuracy and consistency of each method, regarding required criteria defined in this paper. The methods that do not verify these criteria have been ruled out. Finally, the remaining methods have been studied on a benchmark of randomly-generated arrays of mono-dispersed spheroids, with aspect ratios ranging from 1/6 to 6 and volume fractions ranging from 0.05 to 0.25. For each method, the spheroidal packings have been sized to measure a statistically significant computing time. Such procedure enabled to study with generality the computing-time dependency of one method on the aspect ratio, the volume fraction, and the desired accuracy. The most efficient method for a given value of these parameters has then been identified.