The two-sender unicast index coding problem consists of two senders collectively having all the demanded messages of a set of receivers, where each receiver demands a unique message. The senders avail the knowledge of the side-information present at all the receivers to reduce the total number of broadcast transmissions. This problem is relevant in many practical communication problems like multi-source satellite communication, multi-user coded cooperative data exchange, and other related problems. In this paper, the two-sender unicast index coding problem is analyzed using three independent single-sender subproblems. Optimal broadcast rate (total number of transmitted bits per message bit as the message length tends to infinity) for all the unsolved instances of a special class of the two-sender unicast index coding problem is provided in terms of those of the three associated subproblems. The optimal broadcast rate established in this work serves as a lower bound for the optimal broadcast rate of any general associated two-sender unicast index coding problem. An achievable broadcast rate (total number of transmitted bits per message bit) with finite length messages for any finite length, is given for a subclass of the two-sender unicast index coding problem by providing a code-construction. This serves as a tighter upper bound when compared to the prior state of art.