= Microplate fill-height optical cancellation
\b[Microplate fill-height optical cancellation] is the cancellation between dilution and increasing light-path length when a fixed dissolved amount is measured vertically through an ideal parallel-sided <microplate> well. It is a geometric limit of the <Beer-Lambert law>, not a general guarantee that changing liquid volume leaves a <microplate reader> signal unchanged.
Let $n$ be the dissolved amount of <formazan>, $V$ the solution volume, $S$ the constant internal area perpendicular to the beam and $h=V/S$ the liquid height. For a homogeneous solution and a beam traversing that whole height, the background-subtracted <absorbance> is
$$
A-A_0=\epsilon hc=\epsilon\frac{V}{S}\frac{n}{V}=\frac{\epsilon n}{S}.
$$
Here $\epsilon$ is the <molar absorption coefficient>, with units compatible with the units of concentration and length. The <formazan optical response> depends on amount per illuminated well area in this idealization.
For example, increasing volume from $0.10$ to $0.20\,\mathrm{cm^3}$ in a well of area $0.50\,\mathrm{cm^2}$ increases the ideal path from $0.20$ to $0.40\,\mathrm{cm}$. At fixed dissolved amount, concentration halves and path length doubles, leaving their product unchanged. A fixed-path cuvette would instead show half the product <absorbance>. If concentration is held fixed while adding more solution to the well, amount increases and the cancellation argument does not apply.
A <tapered microplate optical path model> shows why well shape matters. Suppose horizontal area at height $z$ is $S(z)=S_0(1+\alpha z)$, with positive area throughout the filled region. Then
$$
V(h)=S_0\left(h+\frac{\alpha h^2}{2}\right),\qquad
A-A_0=\frac{\epsilon n}{S_0(1+\alpha h/2)}.
$$
For an outward taper, increasing fill height now reduces the ideal product <absorbance> at fixed amount. The parallel-sided result is recovered at $\alpha=0$.
This model assumes absorption by dissolved product. A meniscus, an oblique or finite beam, suspended particles, evaporation and instrument-applied <microplate path-length correction> can change the measured response. The manufacturer's <MTT> procedure explicitly separates formation of product crystals, dissolution and multiwell measurement.<Cell Proliferation Kit I MTT instructions by Roche>[Section 3.1 and Figure 4, printed p. 9]{ref} A dilution experiment should therefore specify the optical geometry, product recovery and processing rather than treating volume independence as evidence of complete dissolution or unchanged <cell viability>.
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