If there are higher dimensions, say the extra seven asserted by String Theory, then we have breadth (thickness?) along each axis that is non zero. The higher-order string theory dimensions (which communicate particle information like gravity) are tightly rolled up.

Brian Greene uses the metaphore of an ant on a wire who can move along the wire freely, but can't go far laterally. They may be so small that our quantum bits can't drift anywhere, so our liver doesn't abandon us drift along a high-level axis.

If there are flat higher level dimensions, then either a force or some kind of membrane would have to exist to keep our blood from leaking.

That said, when we have pure elements, or even pure minerals or chemicals, they retain the same density (mass to volume, sometimes affected by temperature) which suggests nothing is hiding away in other dimensions whenever we take measurements. If there is room along higher axes for unseen activity, it doesnt bug us enough to work out consistent properties.