The centrome is a spherical organelle in the cytoplasm of the cell and near the nucleus. It does not have a membrane, instead it is made up of a cloud of proteins called the "centrosome matrix" at the center of which are two centrioles, perpendicular to each other.
Each centriole is made of 9 sets of 3 short microtubules. These microtubules do not break down and rebuild themselves, they stay permanent. The matrix of proteins around them feed the centrioles proteins to allow them to build the microtubules as well as the components of cilia and flagella.