Mass-normalized temporal kernels for nail clippings ($\int w(s)ds=1$)

Mass-normalized temporal kernels for nail clippings assign each earlier material time its fraction of the analyzed sample's mass. They convert a finite clipping into an averaging operator on a time-dependent retained concentration. The weighting is normalized by the assayed material amount, rather than by elapsed time alone.

For an original one-dimensional advection model, let $s$ be the time when a material element crossed a defined anatomical reference, and collect at time $t$. With spatially uniform positive advance speed $v(u)$,Suppose the clipping's retained elements correspond to $s\in [s_1,s_2]$. Let $m(x,t)$ be structural mass per unit length at collection, and assume the marker's concentration per structural mass is preserved along each element. ThenThe kernel has units of inverse time, is nonnegative and integrates to one. It can be treated mathematically like a probability density function for the reference-crossing time of material selected in proportion to its mass. This is a material-accounting interpretation, not a claim that the biological transport is random.

For a stipulated 20-day window, take line density 1 mg/mm, advance 0.10 mm/day in the earlier ten days and 0.20 mm/day in the recent ten days. The two intervals contribute 1 and 2 mg, respectively. Their kernel heights are $1/30$ and $1/15$ per day. A retained concentration of 90 ng/mg only in the earlier interval gives $\overline{c}=30$ ng/mg; a uniform time average would give 45. A constant 30 ng/mg history gives the same sample result. Thus one pooled measurement does not identify a unique history.

A centroid date of a mass-normalized nail kernel summarizes the weighting, while the kernel's spread records its finite time resolution. Neither removes the difference between reference crossing and earlier formation in the hidden nail matrix. The primary nail-marking experiment measures displacement over observation intervals, not the complete kernel or marker retention law.^[ref] Measuring segment mass, geometry and growth history constrains separate parts of this model.

The centroid date of a mass-normalized nail kernel is the weighted mean of the reference-crossing times represented by the analyzed nail material. If $w(s)\ge 0$ is normalized so that $\int w(s)ds=1$, its centroid isAt collection time $t$, the corresponding mean material age is $\overline{a}=t-{\mu }_s$. This summarizes a distribution of material times; it is not automatically the date of an exposure or the instant of formation in the nail matrix.

For an original pooled-sample example, suppose equal structural mass comes from two material-age intervals, 0–2 and 18–20 days before collection. Age density is 0.25 per day in each interval and zero between them. Their interval means are 1 and 19 days, so the mean age is 10 days. Yet no sampled material has age 10 days. A single uniform kernel over 0–20 days has the same mean age, with a very different distribution.

The age variance makes part of the difference explicit. A uniform interval of width 2 days has variance $2^2/12=1/3\mathrm{da}{\mathrm{y}}^2$. The separated mixture therefore haswhereas a uniform 20-day interval has variance $20^2/12=100/3\mathrm{da}{\mathrm{y}}^2$. Equal centroids do not imply equal temporal resolution.

The centroid has a precise averaging property. For any representative time $r$,It minimizes the mass-weighted squared timing discrepancy. Also, if retained concentration is affine in $s$, its kernel average equals its value at ${\mu }_s$. This equality generally fails for a nonlinear history: a quadratic $c(s)=c_0+k(s-{\mu }_s{)}^2$ averages to $c_0+k{\sigma }_s^2$, not $c_0$.

A primary growth experiment followed marks near the proximal nail fold over measured intervals.^[ref] Such displacement observations help constrain the material clock. They do not establish the full kernel, marker retention or one unique exposure date. Report the reference, distributional width and pooling assumptions alongside a centroid date.