where x is the cell count, k is the untreated growth rate (per day), c is the drug concentration, SM is the maximal inhibitory effect, SC50 is the concentration at half-maximal effect of drug, and h the Hill coefficient. The growth rate k corresponds to the division rate k0 as k = ln(2)·k0 = ln(2)/Td, where Td is the division time. SM can be larger than 1 to account for drugs inducing cell death at a specific phase of the cell cycle. The model can also be generalized to account for drugs that induce cell death independent of the cell cycle:
where kL is the maximal killing rate (per day), LC50 is the concentration of drug that produces half-maximal cell killing. Integrating these equations for an assay of t days yields the cell count at concentration c:
where x0 ≡ x(t = 0) is the cell number at the time of treatment. Thus, the relative cell count is:
and the normalized growth rate inhibition (GR value) is:
This equation for GR(c) is independent of the length of the assay t and thus, the metrics GR50, GRmax, GRAUC, and hGR are also independent of t. For cases where drug action is mainly related to the cell cycle (kL =0), GR values are also independent of the untreated growth rate k. As shown in