Mass-transfer Biot number for drug release (${\mathrm{Bi}}_m=hL/D$)

The mass-transfer Biot number for drug releasecompares 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,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. ThenRequiring this last error to be no more than $5\%$ gives ${\mathrm{Bi}}_m\ge 19$. 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 $D=10^{-12}{\mathrm{m}}^2/s$, $L=100\mu \mathrm{m}$ and $h=2\times 10^{-7}m/s$, ${\mathrm{Bi}}_m=20$. 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.^[ref] Such observations motivate checking external resistance; a fitted release curve alone does not measure $h$, $D$ and $L$ independently.

Sink-boundary approximation error in drug release is the difference between a release prediction using an ideal receiving boundary and one that accounts for the actual boundary. The reference prediction and the error denominator must be specified. Finite external resistance and nonzero bulk drug concentration are separate departures from a perfect sink condition in drug release.

For the maintained-reservoir steady planar model, let the internal concentration be $c_i>0$, receiving bulk concentration $c_b$, matrix thickness $L$, internal diffusion coefficient $D$ and external transfer coefficient $h$. With compatible concentration scales,The ideal zero-bulk, negligible-external-resistance prediction is $J_0=D{c}_i/L$. Defining $\beta =c_b/c_i$ and the mass-transfer Biot number for drug release $B=hL/D$ givesThis is the combined boundary-resistance and bulk-concentration error in drug release for the stipulated model. It is not an exact transient error formula for every dosage form.

An original example with $B=20$ and $\beta =0.10$ gives $J/J_0=6/7$. The ideal flux overpredicts by about $14.3\%$ relative to $J_0$, or $16.7\%$ relative to $J$. Making external transport arbitrarily fast still leaves the concentration-related error floor $\beta$ in the first error convention.

The model also supplies a transparent tolerance rule. For $0\le \beta <\epsilon <1$, requiring $(J_0-J)/J_0\le \epsilon$ givesAt $\epsilon =0.05$ and $\beta =0.01$, the threshold is $23.75$. If bulk accumulation already exceeds the tolerance, increasing $h$ alone cannot meet it.

Experimentally, changing receiving volume tests accumulation, while changing flow or membrane transport tests external resistance; sampling correction in drug release testing is needed when material is removed. The FDA’s primary adaptive-perfusion account specifically identifies passive membrane transport as potentially rate-limiting.^[ref] Separating the two effects prevents a large receiving volume from being mistaken for proof of an ideal surface boundary.

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 $J_0=D{c}_i/L$ 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 $0\le \beta \le 1$, with concentrations expressed on compatible scales. ThenWriting the isolated resistance error as $e_R=1/(1+B)$ givesSimply 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 $e/(1-e)\approx 0.1574$.

A useful design calculation solves for how much bulk accumulation is permissible. For a requested tolerance $0<\epsilon <1$,The right side is nonnegative only if $B\ge (1-\epsilon )/\epsilon$. Even zero initial bulk concentration cannot rescue too large an external resistance. At $B=99$ and $\epsilon =0.02$, the permitted bulk fraction is about $0.01010$. For a maintained $c_i=1\mathrm{mg}\mathrm{m}{\mathrm{L}}^{-1}$ 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.^[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.