= Optical contribution of the nail bed
The \b[optical contribution of the nail bed] is the portion of measured light associated with tissue beneath the <nail plate>, after accounting for illumination, the covering plate and collection geometry. The vascular tissue beneath the plate contributes to its ordinary pink appearance, while the compact plate itself consists of dead <keratinocytes>.<Anatomy and Physiology 2e by J. Gordon Betts and others>[§5.2, Nails and Figure 5.13]{ref} A colour measurement combines optical transport and anatomy rather than directly reporting a single <blood> quantity.
An original simplified <reflectance> model is
$$
y=\frac{I}{I_0}=P+T^2R,
$$
where $P$ represents light returned by the plate and interfaces, $T$ is single-pass plate <transmittance> and $R$ is the fraction returned from the underlying region. The square represents outward and return passage through the plate. This deliberately omits detailed <multiple scattering>; it is an explanatory model, not a clinical calibration.
Even with known $P=0$, $y=0.32$ can arise from $(T,R)=(0.8,0.5)$ or $(\sqrt{0.4},0.8)$. One measurement cannot distinguish those pairs. For small changes with $P=0$,
$$
\frac{dy}{y}\simeq2\frac{dT}{T}+\frac{dR}{R}.
$$
Thus a 1% relative change in plate <transmittance> contributes approximately a 2% relative signal change without any change in underlying <reflectance>. This is a concrete <inverse problem>.
A primary experiment separated one optical component physically: thirteen autopsy toenails were measured at fifteen wavelengths between 300 and 600 nm using a <monochromator>, <integrating sphere> and <photodiode>, with a correction for nail <fluorescence>.<The transmission of optical radiation through human nails by S. G. Parker and B. L. Diffey>[Abstract; British Journal of Dermatology 108, 11–16 (1983), DOI 10.1111/j.1365-2133.1983.tb04573.x]{ref} Measuring an isolated plate constrains its transmission; it does not measure living nail-bed <blood flow>.
<Plate-bed optical identifiability> requires additional constraints: independent plate measurements, a specified spectral model or multiple collection geometries. More wavelengths help only if the added data constrain the unknowns; allowing an arbitrary new $T$ and $R$ at every wavelength preserves the ambiguity.
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