= Combined boundary-resistance and bulk-concentration error in drug release
This error combines two departures from an ideal sink prediction: resistance between the releasing surface and receiving liquid, and drug already present in that liquid. For the steady planar reservoir model in <sink-boundary approximation error in drug release>, let $J_0=Dc_i/L$ be the ideal flux, $B=hL/D$ the <mass-transfer Biot number for drug release> and $\beta=c_b/c_i$. Assume $D,L,h,c_i>0$ and $0\leq\beta\leq1$, with concentrations expressed on compatible scales. Then
$$
\frac{J}{J_0}=(1-\beta)\frac B{1+B},\qquad
e=\frac{J_0-J}{J_0}=1-(1-\beta)\frac B{1+B}.
$$
Writing the isolated resistance error as $e_R=1/(1+B)$ gives
$$
e=\beta+e_R-\beta e_R.
$$
Simply adding $\beta$ and $e_R$ counts their overlap twice. For example, $B=9$ and $\beta=0.04$ give $e=0.136$, rather than $0.14$. This error uses the ideal flux as denominator; the overprediction relative to actual flux is $e/(1-e)\approx0.1574$.
A useful design calculation solves for how much bulk accumulation is permissible. For a requested tolerance $0<\varepsilon<1$,
$$
e\leq\varepsilon
\quad\Longleftrightarrow\quad
\beta\leq\varepsilon-\frac{1-\varepsilon}{B}.
$$
The right side is nonnegative only if $B\geq(1-\varepsilon)/\varepsilon$. Even zero initial bulk concentration cannot rescue too large an external resistance. At $B=99$ and $\varepsilon=0.02$, the permitted bulk fraction is about $0.01010$. For a maintained $c_i=1\,\mathrm{mg\,mL^{-1}}$ and a well-mixed receiving volume $100\,\mathrm{mL}$, this corresponds to about $1.01\,\mathrm{mg}$ accumulated in the liquid, assuming no removal or other drug sources. This is an illustrative model calculation, not a general assay acceptance threshold.
Increasing receiving volume primarily limits $\beta$; changing flow or membrane transport can change $B$. The FDA's adaptive-perfusion apparatus specifically replaces potentially rate-limiting passive membrane diffusion with pressure-driven tangential filtration.<Adaptive Perfusion: A Novel In Vitro Drug Release Testing Method for Complex Drug Products by the FDA>[Technology Summary, conventional membrane transport versus pressure-driven separation]{ref} Comparing flow conditions and volumes can distinguish the two limitations. A finite reservoir, changing internal profile, partitioning or sampling requires its own <mass balance> before this steady expression can be interpreted as an error estimate.
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