= Matrix growth and plate-advection balance
\b[Matrix growth and plate-advection balance] relates newly supplied nail material to movement and changing geometry of the <nail plate>. The <nail matrix> contains the growth-producing region, whereas the mature <nail plate> is transported material.<Anatomy and Physiology 2e by J. Gordon Betts and others>[§5.2, Nails: proliferating basal cells at the root]{ref} Measured advance is consequently a material-flow observation, not a direct count of <cell divisions>.
For an original one-dimensional <conservation of mass> model, define line density $m(x,t)=\rho(x,t)A(x,t)$, where $A$ is plate cross-sectional <area>, $\rho$ its <density> and $v$ its advance speed. A local production rate $s(x,t)$ per unit length gives the <continuity equation>
$$
\frac{\partial m}{\partial t}+\frac{\partial(mv)}{\partial x}=s.
$$
If the chosen downstream segment has negligible production or loss, $s=0$. At steady state its <mass> flux $mv$ is constant. Uniform <density> then makes the volume flux $Q=Av$ constant, recovering the simple width-times-thickness model.
For stipulated $Q=0.30\,\mathrm{mm^3/day}$, cross-sectional <areas> of 3 and $2\,\mathrm{mm^2}$ require speeds 0.10 and 0.15 mm/day. Equal production can therefore accompany unequal advance when geometry changes. Conversely, equal speeds do not establish equal production without knowing <area> and <density>.
For a fixed segment $[a,b]$, integrating the <continuity equation> gives
$$
\frac{d}{dt}\int_a^b m\,dx=(mv)(a,t)-(mv)(b,t)+\int_a^b s\,dx.
$$
If inflow exceeds outflow, material accumulates rather than vanishing from the accounting. Growth, hydration-related geometry change and clipping must be assigned to appropriate terms before interpreting an apparent imbalance.
The nail-marking study measured displacement over known time intervals in 22 young adults.<Growth rate of human fingernails and toenails in healthy American young adults by S. Yaemsiri and others>[Abstract, Methods]{ref} Extending that design with repeated width and thickness measurements would test a <nail-plate storage balance>, rather than assuming steady geometry. Inferring a mitotic rate would require further information about cell size and conversion into mature material.
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