= Cell line population doublings
{title2=$d=\log_2(N_1/N_0)$}
For positive recoverable counts $N_0,N_1$ bracketing one growth interval, define its net <cell line population doublings> by
$$
d=\log_2\!\left(\frac{N_1}{N_0}\right)=\frac{\ln(N_1/N_0)}{\ln2}.
$$
The formula counts how many factors of two produce the observed net factor. For exactly two divisions per original <biological cell>, with all daughters recovered, $N_1=2^2N_0$, giving $d=2$. Death or loss of <biological cells> makes this a net estimate of division.<Animal Cell Culture Guide by ATCC>[Passage number and population doubling level, PDL equation and loss caveat]{ref}
Under constant-rate <exponential growth>, $N(t)=N_0e^{rt}$, a growth interval of duration $\Delta t$ gives
$$
r=\frac{\ln(N_1/N_0)}{\Delta t},\qquad t_2=\frac{\ln2}{r}=\frac{\Delta t}{d}.
$$
For the constructed fourfold increase over $48\,\mathrm{h}$, $d=2$ and the model <doubling time> is $24\,\mathrm{h}$. The proof follows by taking the <natural logarithm> and solving $e^{rt_2}=2$. This is a fitted property of that interval and model: it is not a universal <HeLa> <doubling time>, nor the interval between two divisions of each individual <biological cell>. Across successive growth intervals, add their $d$ values; keep transfer losses separate as in <cell culture passage accounting>.
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