Abstract: The computation of blow-up solutions to a differential equation is often a difficult task. Here, we examine a system of ODEs that is derived from the Maxwell-Debye equations. Blow-up times for solutions to the ODE system are estimated using two approaches -- MATLAB event location and a Sundman transformation. The Sundman transformation, whereby a new temporal variable is introduced, results in a system for which solutions exist globally. In addition, it provides a means for simultaneously proving blow-up and finding analytical estimates of blow-up times.