How to find the smallest sphere that intersects 3 given lines in 3D?

Let the three lines be given by parametric equations

$ R_i(t) = P_{i} + t V_{i} ,\hspace{20pt} i = 1, 2, 3 $

where the direction vectors $\{ V_i \}$ are assumed to be unit vectors.

Given the distance $d$ between the center of the sphere $C$ and each of these lines ($d$ is the radius of the sphere), we can write three equations in the vector $C$:

$(C - P_{i} )^T (I_3 - {V_i V_i}^T ) (C - P_{i} ) = d^2, \hspace{20pt} i = 1, 2, 3 $

The above equation(s) are three equations of 3 cylinders (which are quadrics) having axes pointing along $\{V_i \}$ and passing through $\{ P_i \}$ respectively.

I don't know if an analytic solution exists for $C$ given $d$, but what I did was vary $d$ (in a descending variation) and compute $C$ numerically until no solution was possible. This way I was able to determine the minimum $d$ possible which is the radius of the smallest sphere.