Recently, it has been shown that a simple, distributed backlog-based carrier-sense multiple access (CSMA) algorithm is throughput-optimal. However, throughput optimality is established under the perfect or ideal carrier-sensing assumption, i.e., each link can precisely sense the presence of other active links in its neighborhood. In this paper, we investigate the achievable throughput of the CSMA algorithm under imperfect carrier sensing. Through the analysis on both false positive and negative carrier sensing failures, we show that CSMA can achieve an arbitrary fraction of the capacity region if certain access probabilities are set appropriately. To establish this result, we use the perturbation theory of Markov chains.