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.
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