Cell-free tetrazolium interaction contrast

A cell-free tetrazolium interaction contrast isolates the part of an optical change that depends jointly on adding the test compound and adding the tetrazolium salt, in a specified tetrazolium cell-free control. All four mixtures lack cells; the comparison tests chemical or optical interference rather than cellular survival.

Write $R$ for reagent present, $0$ for reagent absent, $T$ for test compound and $V$ for its matched vehicle. From the four absorbance readings defineThe final term restores a baseline that would otherwise be subtracted twice. In the additive modelthe contrast equals the interaction $i$. The terms $r$ and $t$ describe the reagent-only and compound-only contributions relative to the vehicle baseline. This is an algebraic separation under stated conditions, not an assumption that the underlying chemical reactions are always additive.

An original example with readings $0.32,0.12,0.08,0.05$ in that order gives $I=0.17$. Subtracting only $A_{R,V}$ gives $0.24$ and retains the compound-only increment $0.07$. Adding the same constant optical offset to every reading leaves $I$ unchanged, whereas multiplying every reading by an unknown gain also multiplies $I$.

For tetrazolium contrast uncertainty, let the four readings have covariance matrix $\Sigma$ in the order above and put $w=(1,-1,-1,1{)}^{\mathsf{T}}$. Linear uncertainty propagation givesFour independent readings with equal standard uncertainty $u$ give $u_I=2u$. A single shared additive offset contributes zero because the coefficients sum to zero; treating it as four independent offsets would overstate that contribution. Correlations require the appropriate input model.^[ref]

Direct cell-free MTT reduction by selected compounds motivated this kind of control.^[ref] A nonzero contrast establishes a reagent-dependent cell-free optical effect under the matched conditions. Identifying that effect as a particular formazan requires suitable chemical or spectral evidence; transferring its magnitude to a cell-containing sample additionally requires a validated response model.

Tetrazolium contrast uncertainty quantifies uncertainty in the four-reading cell-free tetrazolium interaction contrast, including correlations between its inputs. In the order reagent/test, no-reagent/test, reagent/vehicle and no-reagent/vehicle, writeFor finite input variances and known covariance matrix $\Sigma$, the variance is exactlyThe exactness here follows from linearity of the contrast. JCGM's general uncertainty propagation framework uses sensitivity coefficients and correlated input quantities; this contrast has coefficients $w_i$ independent of the readings.^[ref]

An original example gives each reading an independent residual with standard uncertainty $0.010$ and all four readings the same additive offset with standard uncertainty $0.030$. ThenSince $w^{\mathsf{T}}1=0$, the common offset cancels, giving $u_I=0.020$. Treating those four appearances of one shared offset as independent would instead give $\sqrt{4(0.010^2+0.030^2)}\simeq 0.0632$. The distinction concerns the physical source of variation, not the number of numbers in the subtraction.

A shared multiplicative gain does not cancel. If $A_i=(1+g){\mu }_i+b+e_i$, with independent zero-mean gain error $g$, common offset $b$ and independent residuals $e_i$, thenFor chosen $I_0=0.17$, $u_g=0.05$ and $u_e=0.010$, the combined standard uncertainty is approximately $0.0217$. The gain contribution is a relative uncertainty, whereas the residual contribution is in absorbance units.

The input model should distinguish repeated readings of one preparation from independently prepared controls. A small propagated uncertainty does not test whether controls reproduce the sample's chemistry or whether a nonzero optical contrast identifies a particular formazan. Those are separate questions about the measurement model.