Several elementary heat conduction examples are given demonstrating the need to employ a fictitious initial condition in place of the true initial condition in homogenized, effective medium descriptions of transient thermal phenomena for systems where the boundary conditions are such that the heat originally present in the system is not conserved. The need arises as a consequence of the spatial inhomogeneity of the local rate of heat transfer towards the boundaries during the period existing prior to that at which the effective medium description of the system becomes a faithful macroscale model of the transient micro-scale phenomena. While the illustrations are confined for simplicity to pure conduction problems, the need to use fictitious initial conditions extends to broader classes of problems, including those for which forced convection plays a major role in the thermal transport process.