= Drug release rate
{title2=$r(t)=dM_{\mathrm{rel}}/dt$}
A <drug release rate> is the amount released from a formulation per unit time. If $M_{\mathrm{rel}}(t)$ is cumulative released amount, the instantaneous rate is
$$
r(t)=\frac{dM_{\mathrm{rel}}}{dt}.
$$
Its units are amount per time, such as mass per hour. Dividing by a reference amount gives a fractional rate; dividing by release area gives a <mass flux>. These normalizations answer different questions and should be stated explicitly.
In discrete measurements, one commonly obtains an interval mean:
$$
\bar r_{12}=\frac{M_{\mathrm{rel}}(t_2)-M_{\mathrm{rel}}(t_1)}{t_2-t_1}
=\frac1{t_2-t_1}\int_{t_1}^{t_2}r(t)\,dt.
$$
For illustrative cumulative amounts $0$, $2$ and $3\,\mathrm{mg}$ at zero, one and three hours, the interval means are $2$ and $0.5\,\mathrm{mg\,h^{-1}}$. The second value is an average over two hours, not necessarily the instantaneous rate at their midpoint. A decreasing rate can coexist with a cumulative amount that continues to increase.
Differentiation also amplifies measurement noise. With times treated as exact, <propagation of uncertainty> gives
$$
\operatorname{Var}(\bar r_{12})=
\frac{\operatorname{Var}(M_2)+\operatorname{Var}(M_1)-2\operatorname{Cov}(M_1,M_2)}{(t_2-t_1)^2}.
$$
For independent amount errors with standard deviation $0.10\,\mathrm{mg}$ each, a one-hour interval gives rate standard deviation about $0.141\,\mathrm{mg\,h^{-1}}$; shortening the interval to $0.10\,\mathrm h$ raises it to about $1.41\,\mathrm{mg\,h^{-1}}$. Repeated-sampling corrections can introduce shared errors, so independence must not be assumed automatically.
An <in vitro drug release test> measures release through an apparatus, and the separation step can influence the rate observed.<Adaptive Perfusion: A Novel In Vitro Drug Release Testing Method for Complex Drug Products by the FDA>[Technology Summary]{ref} Reporting the amount definition, normalization, time interval, <sampling correction in drug release testing> and uncertainty makes a rate estimate interpretable. A smoother fitted derivative can be useful, but it also introduces assumptions about the curve between observations.
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