Microtubules (MTs) are protein filaments found in all eukaryotic cells which are crucial for many cellular processes including cell movement, cell differentiation, and cell division. Due to their role in cell division, they are often used as targets for chemotherapy drugs used in cancer treatment. Experimental studies of MT dynamics have played an important role in the development and administration of many novel cancer drugs, however, a complete description of MT dynamics is lacking. Here, we propose a new mathematical model for MT dynamics, that can be used to study the effects of chemotherapy drugs on MT dynamics. Our model consists of a growth-fragmentation equation describing the dynamics of a length distribution of MTs, coupled with two ODEs that describe the dynamics of free GTP-and GDP-tubulin concentrations (the individual dimers that comprise of MTs). Here, we prove the well-posedness of our system and perform a numerical exploration of the influence of certain model parameters on the systems dynamics. In particular, we focus on a qualitative description for how a certain class of destabilizing drugs, the vinca alkaloids, alter MT dynamics. Through variation of certain model parameters which we know are altered by these drugs, we make comparisons between simulation results and what is observed in in vitro studies.