MTBF is normally expressed as a failure rate per specific time, usually per every thousand flying hours.
You can either use a know failure number or if this is not available a generic number. This can be a high failure number for early failures such as infant mortality, a lower random failure number for the bottom of the bathtub shape and a higher age related or wear out number.
Next is the likelihood of such a failure occurrence per unit of time. If you assume a typical unit of time is one flying hour then you need to consider (in this case) takeoff. If you assume the takeoff phase as two minutes, which would include the climb, then MTBF can be calculated.
As an example the engine failure characteristic is one per thousand flying hours then the MTBF just for the takeoff phase would equate to one thirtieth of one per thousand flying hours, assuming a single engine case.
This is a fairly simplistic case as to be more accurate you could consider that the engine might be at maximum power/maximum stress during that two minutes. Having said that the engine should be designed for the maximum power situation.
I hope that this helps.