Trilinear interpolation for distorted cubes?
Imagine a cube with each corner a different RGB color value - as you know you could interpolate the colors inside the cube using trilinear interpolation. 

You can reuse this same logic for coordinates as well -- set the RGB values of the corners to be the XYZ coords of your distorted cube's corners. 

Now the RGB "color" you sample in this cube is the distorted coordinate -- eg, it is the coordinate of your sampling point after that point has been "remapped" to the now-distorted cube, using the normal trilinear algorithm.

Hope this helps.