Word for a poem in which lines begin with letters of a previous word

I was wondering if someone knew the word that describes a poem where each word begins with the letters of a previous word, of if such a word even exists.

An example of such a poem is this: http://crosswords.net23.net/poems/Joshua.html (I wrote it)

Someone told me there was a word for this kind of thing, but he couldn't remember it, either.

Solution 1:

You started with an acrostic of 'JOSHUA':

An acrostic is a poem or other form of writing in which the first letter, syllable or word of each line, paragraph or other recurring feature in the text spells out a word or a message.

Each instance after that is new acrostic of the word made from each letter in the original.

Solution 2:

The scholarly term technopaegnia encompasses all kinds of interplay between the words of a poem and its structure, including (but not limited to) picture poems, acrostics, and other such puzzles. For facts about the term and links to many interesting examples of such poetry – ancient and modern – see my answer to another question.