The present study analyzes the optimal distribution of a limited amount of high thermal conductivity material to enhance the heat removal of circular 3D integrated circuits, IC. The structure of the heat spreader is designed as a composite of high thermal conductivity (Boron Arsenide) and moderate thermal conductivity (copper) materials. The volume ratio of high-conductivity inserts to the total volume of the spreader is set at a fixed pertinent ratio. Two different boundary conditions of constant and variable temperature are considered for the heat sink. To examine the impact of adding high-conductivity inserts on the cooling performance of the heat spreader, various patterns of the single and double ring inserts are studied. A parametric study is performed to find the optimal location of the rings. Moreover, the optimal distribution of the high-conductivity material between the inner and outer rings is found. The results show that for the optimal conditions, the maximum temperature of the 3D IC is reduced up to 10%; while the size of the heat sink, and heat spreader can be diminished by as much as 200%.