  The inflated graph $G_{I}$ of a graph $G$ with $n(G)$ vertices is obtained
from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is
isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is
replaced by an edge $(u,v)$ in such a way that $u\in X_{i}$, $v\in X_{j}$, and
two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. For
integer $k\geq 1$, the $k$-tuple total domination number $\gamma_{\times
k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set
of $G$, which is a set of vertices in $G$ such that every vertex of $G$ is
adjacent to at least $k$ vertices in it. For existing this number, must the
minimum degree of $G$ is at least $k$. Here, we study the $k$-tuple total
domination number in inflated graphs when $k\geq 2$. First we prove that
$n(G)k\leq \gamma_{\times k,t}(G_{I})\leq n(G)(k+1)-1$, and then we
characterize graphs $G$ that the $k$-tuple total domination number number of
$G_I$ is $n(G)k$ or $n(G)k+1$. Then we find bounds for this number in the
inflated graph $G_I$, when $G$ has a cut-edge $e$ or cut-vertex $v$, in terms
on the $k$-tuple total domination number of the inflated graphs of the
components of $G-e$ or $v$-components of $G-v$, respectively. Finally, we
calculate this number in the inflated graphs that have obtained by some of the
known graphs.
