How to make grow a geometry

[mahesh-kumar--8]

Hi i'm working in production in a studio, i've had critical problem in my project to growing a geometry. that geometry wanna to twists animatedly in a model. but i don't know how to make it. i've attached a screenshot of it.  It represents a tiny model in it's left side it should wanna to grow  along a curve or something plz help

 thanks in advance. 

[bjohnston]

You could extrude along a curve.