Why does duty cycle seem relevant when calculating inductor current in boost converter circuit?
Stop! You will need some incredible luck if you are going to get something usable by shuffling formulas without understanding them.

Start by learnig how inductors actually work. You must both qualitatively and also in numbers understand how a coil stores magnetic energy, when its current gradually grows after DC voltage is applied to it. You should also understand that the current never stops immediately, but gradually, the coil will generate as high voltage as needed to let the current flow despite you turned the switch off. Start by reading this:

How does the inductor &#39;&#39;really&#39;&#39; induce voltage?

Then learn the induction law and try to calculate some current growth rates with known dc voltage and inductance. Calculate also how inductor current decreases, when the generated inductive peak is supplied to known existing DC voltage. 

Then you can learn how flyback type switchmode  power supplies really work. The duty cycle is a trivial and natural quantity in repeating pulse systems to describe how big part of the time something is on, typically the switching transistor.