Tapered microplate optical path model

A tapered microplate optical path model accounts for a well's changing horizontal area when relating liquid volume to the vertical light path. The ideal microplate fill-height optical cancellation is exact for parallel sides; a tapered well generally breaks that cancellation even for a fully dissolved, homogeneous dye.

Let the internal area at height $z$ be $S(z)$ and the filled height be $h$. For dissolved amount $n$ distributed uniformly in volume $V(h)$,The last equality uses the Beer-Lambert law for an axial beam that traverses the whole liquid height. Thus the relevant area is the height-averaged area $V(h)/h$, not automatically the bottom area or the area at the meniscus.

For an original area-linear model $S(z)=S_0(1+\alpha z)$, with positive area throughout the liquid,For outward taper $\alpha >0$, raising the fill height at fixed dissolved amount lowers the ideal absorbance. With stipulated $S_0=0.50\mathrm{c}{\mathrm{m}}^2$ and $\alpha =1\mathrm{c}{\mathrm{m}}^{-1}$, heights $0.20$ and $0.40\mathrm{cm}$ correspond to volumes $0.110$ and $0.240\mathrm{c}{\mathrm{m}}^3$. The signal ratio is $1.10/1.20=11/12$, despite unchanged product amount. These are geometric examples, not dimensions assigned to a commercial plate.

Area increasing linearly with height is not an exact conical frustum. If the radius instead follows $r(z)=r_0+\beta z$, integration givesBoth models recover the constant-area limit when taper vanishes. They assume the same molar absorption coefficient and full product recovery at every volume. The manufacturer separates MTT product formation from dissolution and optical measurement.^[ref] Suspended particles, a curved meniscus, an oblique beam or a reader's microplate path-length correction need additional treatment. A change in signal with dilution therefore does not, by itself, establish a change in formazan amount or cell viability.