Osculating sphere related rates

Given the vector function $\vec{r}(t) = \left \langle 2t, t^2, \dfrac{t^3}{3} \right \rangle$ , you can get an osculating circle to $\vec{r}(t)$ at any point in time $t$ . Use this osculating circle as the largest trace of a sphere (so now when $t$ changes, you have a sphere moving along the curve with its largest trace as the osculating circle to $\vec{r}(t)$ ). If the rate of change of the volume of this sphere with respect to $t$ at $t=1$ can be represented in the form $A \pi$ , find $A$ .

Clarification:

The osculating circle to $\vec{r}(t)$ at time $t$ is the circle tangent to $\vec{r}(t)$ at time $t$ with radius $\dfrac{1}{\kappa (t)}$ , where $\kappa(t)$ is the curvature of $\vec{r}(t)$ at time $t$ .