= Mass-transfer Biot number for drug release
{title2=$\mathrm{Bi}_m=hL/D$}
The <mass-transfer Biot number for drug release>
$$
\mathrm{Bi}_m=\frac{hL}{D}
$$
compares internal diffusive resistance with external mass-transfer resistance in a specified release model. Here $h$ is a receiving-side transfer coefficient, $L$ the chosen matrix transport length and $D$ the internal <diffusion coefficient>. Its value is dimensionless. The definition requires compatible concentration scales; interfacial partitioning must be included if the phases do not share one.
For the steady planar model in <finite mass-transfer boundary in drug release>,
$$
J=\frac{c_i-c_b}{L/D+1/h},\qquad
\frac{L/D}{1/h}=\mathrm{Bi}_m.
$$
A large $\mathrm{Bi}_m$ means external resistance is small relative to the specified internal resistance. A small $\mathrm{Bi}_m$ means receiving-side transport can dominate. The comparison says nothing by itself about whether the bulk concentration $c_b$ is near zero, whether $D$ is constant or whether the chosen steady model applies.
The <sink-boundary approximation error in drug release> can be quantified in this model. With fixed $c_i-c_b$, let $J_\infty=D(c_i-c_b)/L$ be the negligible-external-resistance flux. Then
$$
\frac{J}{J_\infty}=\frac{\mathrm{Bi}_m}{1+\mathrm{Bi}_m},\qquad
\frac{J_\infty-J}{J_\infty}=\frac1{1+\mathrm{Bi}_m}.
$$
Requiring this last error to be no more than $5\%$ gives $\mathrm{Bi}_m\ge19$. If the error is instead divided by the actual finite-resistance flux $J$, the ratio is $1/\mathrm{Bi}_m$ and the corresponding threshold is $20$. A numerical rule is incomplete without its denominator and model assumptions.
For an original example with $D=10^{-12}\,\mathrm{m^2/s}$, $L=100\,\mathrm{\mu m}$ and $h=2\times10^{-7}\,\mathrm{m/s}$, $\mathrm{Bi}_m=20$. The external-resistance-free approximation exceeds the model flux by $5\%$ relative to $J$, or about $4.76\%$ relative to $J_\infty$.
A release experiment can change receiving flow or membrane transport while keeping the formulation fixed. The FDA’s adaptive-perfusion account specifically contrasts passive dialysis-membrane diffusion with pressure-driven separation.<Adaptive Perfusion: A Novel In Vitro Drug Release Testing Method for Complex Drug Products by the FDA>[Technology Summary, rate-limiting membrane diffusion and tangential flow filtration]{ref} Such observations motivate checking external resistance; a fitted release curve alone does not measure $h$, $D$ and $L$ independently.
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