How to fill in spaces between voronoi holes?

Question

I'm trying to model a scene with bone tissue, like in this pic:

I've gotten this far, but I need those nice, rounded holes like in the pic above.

I've ALMOST achieved this effect by going into edit mode, selecting the whole thing, and hitting "Smooth" in the Transform toolbar on the left a few times. But it deforms the part where the bones meet up with a space, as shown here:

How do I get rid of those spaces and make the bones join up smoothly?

Edit: Also, here's the blend file for the first screen shot:

Answer 1

Check this blend here:

For geometry like this, I often use the Skin Modifier, because it can create these smooth branches automatically. But to use it, you need a mesh as a base, not a bunch of curves. So first, remove the curve profile from your geometry:

Next, use Alt + C to convert the lines to a mesh geometry:

In Edit mode, remove doubles, using a high threshold value, to merge as many vertices as possible while still maintaining the shape you created initially. I was using a merge distance of 0.1 for your scene:

next, add the skin modifier (can take a while until it is on there, and save your scene before):

Finally, go to Edit Mode, and select all vertices, then hit Ctrl + A. This activates a modal operator, when you now drag the mouse you can scale the profile up and down. Scale down until the branch thickness is what you like, and then add a Subsurf Modifier to smooth out the skinned mesh:

You can scale each point individually using the shortcut I mentioned before, so variation can be created easily. You might however get some areas with artifacts. Sometimes, marking a different vertex as the root can resolve or at least improve it:

Now play with the thickness:

Answer 2

Here's an alternative and easy way, based solely on modifiers: Start with a simple cube. Add a subD modifier, and give it 5 iterations to create a cubic sphere. Next, apply a Cast modifier to make it perfectly round (just use default settings). Then add a Decimate modifier using the Planar option, and give it 3 iterations (based on the 5 iterations of your previous SubD modifier). Change the Angle Limit to 20, check ALL BOUNDARIES (very important), and make sure Delimit is set to Normal.

At this stage, with all the modifiers in the stack, your cube should appear to be a Voronoi sphere (or roughly spherical in shape). Add another Cast modifier with default settings just to get it as spherical as it can with the given edges currently on the object. Now add a Wireframe modifier. Set thickness to 0.1 or so. Finally, add another SubD modifier and give it a couple iterations until you're happy with the results. Done.