Consumer surplus in case of perfectly inelastic demand How do we define consumer surplus in the case of perfectly inelastic demand?

This question was inspired by the comments following this answer. For a graph of inelastic demand please also see the linked question.

A motivation for a definition would be that while consumer surplus would be strange in this special case, change in consumer surplus may still be measurable easily with quite reasonable definitions.
From a purely theoretical perspective, if an individual's demand curve is perfectly inelastic, then her willingness to pay for the good is infinite. NB this also implies that she has an infinite budget. Thus, consumer surplus is well defined: it is the willingness to pay minus the price she pays, so as long as the price is finite her consumer surplus is finite.

In practice, no one has an infinite budget. So if the individual's demand curve is truly perfectly inelastic (i.e. the inverse demand is vertical), there exists a price such that beyond that price she can no longer afford to buy the good. This price is her willingness to pay, so consumer surplus is again well defined: the willingness to pay minus the price.