Chapter 2: Signal-to-Concentration Translation and Its Statistical Limitations

In common ADA assay practice, the Screening Cut Point (SCP)—originally determined on the signal scale—is later expressed as a concentration value using the dilution curve from the positive control (PC). 

“…During development, sensitivity may be assessed by testing serial dilutions of a positive control antibody of known concentration, using individual or pooled matrix from treatment-naïve subjects. The dilution series should be no greater than two- or threefold, and a minimum of five dilutions should be tested. The sensitivity can be calculated by interpolating the linear portion of the dilution curve to the assay cut-point…”

— U.S. FDA, Immunogenicity Testing of Therapeutic Protein Products (2019)

In most cases, the SCP lies between two adjacent PC dilutions. Connecting those two points yields a straight line that defines the local signal–concentration mapping:

\[ S = \beta_0 + \beta_1 C \]

where

\[ \beta_1 = \frac{S_2 - S_1}{C_2 - C_1}, \qquad \beta_0 = S_1 - \beta_1 C_1. \]

This is not a statistical regression in the conventional sense—it is linear interpolation between two points. Mathematically, however, it is identical to the ordinary least-squares (OLS) regression fitted to exactly two observations.

Proof: Two-Point Regression ≡ Linear Interpolation

Let the data be \((C_1,S_1)\) and \((C_2,S_2)\) with \(C_1 \ne C_2\).

OLS coefficients are

\[ \hat\beta_1 = \frac{\sum (C_i-\bar C)(S_i-\bar S)}{\sum (C_i-\bar C)^2} = \frac{S_2 - S_1}{C_2 - C_1}, \quad \hat\beta_0 = \bar S - \hat\beta_1 \bar C = S_1 - \hat\beta_1 C_1. \]

Substituting back gives

\[ S(C) = S_1 + \frac{S_2 - S_1}{C_2 - C_1}\,(C - C_1), \]

which is exactly the interpolation formula.
Because both points lie on this line, residuals are zero and no separate linearity test can be defined.

Inverse Mapping

The corresponding concentration for any target signal \(S_\ast\) is obtained by inversion:

\[ C_\ast = C_1 + \frac{S_\ast - S_1}{S_2 - S_1}\,(C_2 - C_1) = \frac{S_\ast - \beta_0}{\beta_1}. \]

Thus, the apparent “conversion of SCP signal to concentration” is a deterministic geometric operation, not a statistical estimation.

Practical Implication

  • The slope (β₁) and intercept (β₀) are geometric parameters connecting two known dilution points.
  • No independent linearity assessment is applicable, since two points always define a unique line.
  • Reported concentration values derived this way should therefore be understood as interpolated representations of the original signal domain.
  • The true 1 % LPC-failure probability remains defined in the signal scale, consistent with the FDA principle:

“…lead to the rejection of an assay run 1 % of the time.”

Note on Directionality and % Inhibition Scale

The interpolation assumes a monotonic relationship between signal and concentration within the local range surrounding the SCP.

  • In bridging ADA assays, the signal increases with analyte concentration (\(S_2 > S_1, C_2 > C_1\)), producing a positive slope.
  • In competitive or inhibition-type assays, the signal decreases with increasing analyte concentration (\(S_2 < S_1, C_2 > C_1\)), leading to a negative slope.

Despite this change in direction, the same linear interpolation principle applies because the function remains monotonic.

In confirmation assays, responses are typically expressed as percent inhibition (%Inhibition) instead of raw signal, defined as:

\[ \%Inhibition = 100 \times \left(1 - \frac{S_{\text{drug-spiked}}}{S_{\text{drug-unspiked}}}\right). \]

Here, \(S_{\text{drug-spiked}}\) is the mean signal measured in wells containing excess drug,
and \(S_{\text{drug-unspiked}}\) is the mean signal from unspiked (screening) wells.
This transformation simply reverses the signal axis while preserving a linear and monotonic mapping.
Therefore, the same local linear interpolation principle applies to concentration translation in both screening and confirmation formats.

Summary

  • The common practice of expressing SCP and LPC in concentration units is based on two-point linear interpolation, mathematically identical to a 2-point OLS regression.
  • Because the interpolation passes exactly through both dilution points, linearity testing is unnecessary in this local range.
  • The statistical design (1 % failure probability) must always be referenced to the signal domain; concentration values merely provide a convenient reporting scale.