= Spatial weighting of nail-clipping exposure records
\b[Spatial weighting of nail-clipping exposure records] determines how much each region of a clipping contributes to a pooled chemical measurement. A concentration reported per unit nail <mass> is a mass-weighted average. Equal lengths of <nail plate> need not supply equal amounts of material if width, thickness or <density> differs.
Let $m(x)$ be <mass> per unit length at collection and $c(x)$ the retained concentration per unit material <mass>. For a clipping spanning $[x_1,x_2]$, an original ideal recovery model gives
$$
M=\int_{x_1}^{x_2}m(x)\,dx,\qquad
\overline c=\frac1M\int_{x_1}^{x_2}m(x)c(x)\,dx.
$$
This assumes complete, composition-independent recovery by the assay. If recovery varies spatially, the measurement equation requires that additional weighting.
For two equal-length regions with masses 1 and 3 mg and stipulated concentrations 0 and 100 units per mg, the pooled concentration is $(1(0)+3(100))/(1+3)=75$ units per mg. A length-weighted average would give 50 and answer a different question. The numerical example illustrates geometry and normalization, not an observed exposure.
<Mass-normalized temporal kernels for nail clippings> arise when positions are converted into earlier reference-crossing times. If $x=x(s)$ is the material map at collection, the normalized time weighting is
$$
w(s)=\frac{m(x(s))\lvert dx/ds\rvert}{M},\qquad
\overline c=\int w(s)c(s)\,ds.
$$
At constant advance speed and uniform line density, the kernel is rectangular, recovering <temporal averaging in nail clippings>. Variable material geometry or speed changes its shape. A growth experiment following file marks measures displacement over a known interval, not this entire weighting function.<Growth rate of human fingernails and toenails in healthy American young adults by S. Yaemsiri and others>[Abstract, Methods]{ref}
A practical study retains segment orientation and measures the material amount used in each assay. Pooling several digits introduces further weights and potentially different growth histories. Wet-mass and structural-mass normalization also differ when water content changes, connecting the issue to <hydration versus production in nail mass balance>. Specify the denominator and material-to-time map before assigning a pooled concentration to one exposure date.
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