= Water-denominator effects in nail concentration measurements
{title2=$C_{\rm wet}=A/(M_s+W)$}
\b[<Water-denominator effects in nail concentration measurements>] occur when a concentration changes because the amount of retained <water> changes the sample <mass> used for normalization. This can happen without any change in the amount of the measured substance. The denominator is part of the reported observable.
For an original inventory model, let $A$ be the retained marker amount, $M_s$ structural <mass> and $W$ water <mass>. Define the water fraction of wet material as $f=W/(M_s+W)$. Concentrations on the two stated bases obey
$$
C_{\rm wet}=\frac{A}{M_s+W},\qquad C_s=\frac A{M_s},\qquad C_{\rm wet}=(1-f)C_s.
$$
At fixed $A$ and $M_s$, increasing hydration lowers $C_{\rm wet}$ while leaving $C_s$ unchanged. The structural basis here is a model-defined material amount; an actual drying procedure must establish what it retains before being equated with $M_s$.
For stipulated $A=100$ ng and $M_s=10$ mg, increasing $W$ from 2 to 5 mg changes the wet-basis concentration from $100/12=8.33$ to $100/15=6.67$ ng/mg, a 20% decrease. Both concentrations correspond to 10 ng/mg on the structural basis. Comparing the wet values alone would falsely suggest a marker loss if unchanged hydration were assumed.
The relation also affects pooled samples. Take two clippings with equal structural <mass> 10 mg and equal marker amount 100 ng, but water amounts 0 and 10 mg. Their wet-basis concentrations are 10 and 5 ng/mg. Pooling gives $200/30=6.67$ ng/mg, rather than the unweighted average 7.5. <Mass-normalized temporal kernels for nail clippings> must therefore use the same mass basis as the constituent concentrations.
A further assay factor can change the numerator as well. If a fraction $q$ of the retained marker is recovered, the measured quantity is $qA/(M_s+W)$. Correcting water content cannot by itself establish $q=1$ or prove that $A$ stayed constant. An original comparison should independently track conditioning, sample <mass> and recovery before attributing a concentration difference to changed incorporation.
A primary <ellipsometry> experiment observed the <complex refractive index> of fingernails during hydration and dehydration, relating the response to water associated with <keratin>.<Hydration dynamics of human fingernails: An ellipsometric study by B. Schulz and others>[Abstract, changing optical response and water interpretation]{ref} This supports treating hydration as a material-state variable. The denominator equations follow from <conservation of mass>; the optical experiment does not directly measure marker retention or matrix production.
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