Please explain the algebraic reasoning in this textbook

Question

I am reading Dennis M. Brown “Drug Delivery Systems in Cancer”. On Page 29 when referring to monotonic increase or decrease of n (being the number of cells n(t) I believe) then how are they getting

AUC₅₀ = T^(1-1/n)

I get how it equal k^(1/n) because we just raise each side of equation IC₅₀ⁿ * T = k to the 1/n power but not getting the the double equality here.

The relationship between IC₅₀ and AUC₅₀ occurs when n = 1 in (IC₅₀)ⁿ * T = k which reduces to AUC₅₀ = k, meaning effect only depends on AUC.

Answer 1

Let's take the general equation they mentioned, (IC₅₀)ⁿT = k, and rearrange it to IC₅₀ = (k/T)^1/n = k^1/nT^-1/n. (IC₅₀)ⁿT = k reducing to AUC₅₀ = k if n=1 implies that AUC₅₀ = IC₅₀T if n=1. If this equation holds for other n, then AUC₅₀ = k^1/nT^1-1/n. That looks a LOT like the text if the author accidentally inserted a =.

This next bit is more speculative because the exact descriptions of these terms in the source are not available online, but AUC generally stands for "area under curve," aka integration. I have doubts AUC₅₀ is some definite integral of IC₅₀ because when n=1, IC₅₀ = k/T, which has an indefinite integral F(T) = k*ln(T) + constant.