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Calculate bounding sphere based on minimum screen size

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Hey Guys, I have (at least for me) a bit of a tough time figuring something out. I want to create a bounding sphere around an object, based on the minimum size it has to have on the screen. I have all camera parameters at hand, screen size at hand and the size of the bounding volume of the object itself, i just need to somehow calculate the distance towards the object in world space where the object starts to come into the field of possibly beeing 10% of screen size. Of course if the object is rotated somehow the area might be less then 10% even though the area is calculated right, but when the viewer comes into that area i can spend more expensive calculations. I made an image that might give some more visual idea to the "problem", and i was hoping someone might give me some hints, tips or solutions to the problem... Thanks in advance

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