This paper studies the performance improvement of the IEEE 802.15.4 nonbeacon-enabled mode originated by the inclusion of the Request-To-Send/Clear-To-Send (RTS/CTS) handshake mechanism resulting in frame concatenation. Under IEEE 802.15.4 employing RTS/CTS, the backoff procedure is not repeated for each data frame sent but only for each RTS/CTS set. The maximum throughput and minimum delay performance are mathematically derived for both the Chirp Spread Spectrum and Direct Sequence Spread Spectrum Physical layers for the 2.4 GHz band. Results show that the utilization of RTS/CTS significantly enhances the performance of IEEE 802.15.4 applied to healthcare in terms of bandwidth efficiency.