In this contribution, we study the total { k }-domination number on graphs. We establish a general upper bound for this number and provide sufficient conditions on a graph to satisfy it at equality. Moreover, for the family of caterpillar graphs, this bound is also tight. The total { k }-domination problem consists of finding a function of minimum value, defined on a set of vertices in a graph, such that in any open neighborhood it has value at least k. We focus on this problem in caterpillar graphs and show that, on this family, it can be reduced to the usual total domination problem (k = 1). Then, we present a representation of a caterpillar in terms of the number of vertices of degree 3 (parents) in it, and the length of the paths induced between two consecutive parents in the central path of the caterpillar. Using this representation, we establish the main result of this work: the value of the total { k }-domination number of every caterpillar, for all k .