The detection tester

Three small tools of molecular diagnosis and gene therapy, written as functions. The first counts the cycles of PCR, the second reads an ELISA plate, and the third follows the engineered cells of a patient.

The idea. PCR doubles the copies in each cycle. In ELISA, a serum is diluted by half in each well, and the last well with a colour gives the titre. The engineered cells halve in number in every half-life.

Your tasks.

1. Finish cycles_needed. It gives the smallest number of cycles that brings start copies to at least needed copies.
2. Finish titre. It gives the dilution of the last well with a reading of at least cut_off. The first well is a dilution of 2, and each well doubles it.
3. Finish level_after. It gives the per cent of the engineered cells that is left after months, for a given half_life.