  Chiral Perturbation Theory predicts the lifetime of pionium, a hydrogen-like
$\pi^+ \pi^-$ atom, to better than 3% precision. The goal of the DIRAC
experiment at CERN is to obtain and check this value experimentally by
measuring the break-up probability of pionium in a target. In order to
accurately measure the lifetime one needs to know the relationship between the
break-up probability and lifetime to a 1% accuracy. We have obtained this
dependence by modeling the evolution of pionic atoms in the target using Monte
Carlo methods. The model relies on the computation of the pionium--target atom
interaction cross sections. Three different sets of pionium--target cross
sections with varying degrees of complexity were used: from the simplest first
order Born approximation involving only the electrostatic interaction to a more
advanced approach taking into account multi-photon exchanges and relativistic
effects. We conclude that in order to obtain the pionium lifetime to 1%
accuracy from the break-up probability, the pionium--target cross sections must
be known with the same accuracy for the low excited bound states of the pionic
atom. This result has been achieved, for low $Z$ targets, with the two most
precise cross section sets. For large $Z$ targets only the set accounting for
multiphoton exchange satisfies the condition.
