Nail-plate storage balance (source code)

= Nail-plate storage balance

\b[Nail-plate storage balance] accounts for material accumulating in a chosen <nail plate> segment when production and removal are unequal. A measured advance speed is a boundary-transport observation; it does not alone specify how much material is stored or produced.

For an original fixed-segment <mass conservation> model, let $M(t)$ be stored structural <mass>, $F_{\rm in}$ and $F_{\rm out}$ its boundary fluxes, $P$ local production and $L$ other losses. Then
$$
\frac{dM}{dt}=F_{\rm in}-F_{\rm out}+P-L.
$$
This is the integrated <continuity equation>. If production is confined outside the selected segment, $P=0$ there even though the upstream <nail matrix> continues producing material. Moving the measurement boundary changes which processes appear as local production or inflow.

With constant <density>, an illustrative volume version has inflow $0.30\,\mathrm{mm^3/day}$ and outflow $0.24\,\mathrm{mm^3/day}$, giving storage gain $0.06\,\mathrm{mm^3/day}$. After ten days the gain is $0.60\,\mathrm{mm^3}$. Spread over a fixed $10\times20\,\mathrm{mm^2}$ footprint, that corresponds to an average thickness increase of 0.003 mm. These are geometric consequences of the chosen model, not measured nail-growth constants.

Clipping is a discrete loss. If a clipping of <mass> $m_c$ is removed at time $t_c$, the balance includes $M(t_c^+)=M(t_c^-)-m_c$. A smooth growth curve fitted across this jump could misinterpret removal as decreased production. Repeated position measurements should therefore be accompanied by clipping records.

<Hydration versus production in nail mass balance> separates structural material from <water>: $M_{\rm wet}=M_{\rm structural}+M_{\rm water}$. A rise in wet <mass> need not mean new structural material. A primary <ellipsometry> experiment measured changes in the <complex refractive index> of human fingernails during water uptake and loss, using a model with more weakly and more tightly associated water.<Hydration dynamics of human fingernails: An ellipsometric study by B. Schulz and others>[Abstract; Physical Review E 65, 061913 (2002), DOI 10.1103/PhysRevE.65.061913]{ref} That establishes a conditioning-dependent material response, not matrix proliferation.

A useful experiment measures marked displacement, segment geometry and <mass> under matched conditioning while recording removals. The measurement model must identify whether it tracks wet material, structural material or an optical proxy; these are different storage quantities.