An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor receive distinct colors. A graph G is said to be injectively $k$-choosable if any list $L(v)$ of size at least $k$ for every vertex $v$ allows an injective coloring ${phi}(v)$ such that ${phi}(v){in}L(v)$ for every $v{in}V(G)$. The least $k$ for which G is injectively $k$-choosable is the injective choosability number of G, denoted by ${chi}^l_i(G)$. In this paper, we obtain new sufficient conditions to be ${chi}^l_i(G)={Delta}(G)$. Maximum average degree, mad(G), is defined by mad(G) = max{2e(H)/n(H) : H is a subgraph of G}. We prove that if mad(G) < $frac{8k-3}{3k}$, then ${chi}^l_i(G)={Delta}(G)$ where $k={Delta}(G)$ and ${Delta}(G){geq}6$. In addition, when ${Delta}(G)=5$ we prove that ${chi}^l_i(G)={Delta}(G)$ if mad(G) < $frac{17}{7}$, and when ${Delta}(G)=4$ we prove that ${chi}^l_i(G)={Delta}(G)$ if mad(G) < $frac{7}{3}$. These results generalize some of previous results in [1, 4].