Antigen Heterogeneity in Dual-Target Engagement

Why Tumor Heterogeneity Changes the Coverage Calculation

A dual-target agent may recognize two tumor antigens, but that does not mean every tumor cell expresses both, or either. A tumor can contain cells expressing antigen A only, antigen B only, both antigens, or neither. These groups matter because the fraction of cells that can be reached is not necessarily the fraction that will be killed. Coverage describes which cells meet a stated antigen criterion; predicted killing also depends on how effectively treatment acts on each group.

The useful starting point is therefore a mixture representation. It makes overlap explicit, avoids treating two antigen prevalences as if they belonged to separate populations, and allows functional response to be calculated for each subpopulation before being combined.

Coverage Is a Set-Overlap Quantity

If a single-target treatment can reach cells positive for its target, A-target coverage is \(C_A=f_{10}+f_{11}\), while B-target coverage is \(C_B=f_{01}+f_{11}\). If a dual-target treatment can reach a cell when either target is present, its union coverage is

The subtraction of \(f_{11}\) prevents dual-positive cells from being counted twice. If the specified mechanism instead requires both target antigens on the same cell, the relevant coverage is \(C_{\mathrm{both}}=f_{11}\), not \(C_{\mathrm{either}}\). Thus, “dual-target coverage” is incomplete unless the criterion—either target or both targets—is stated.

These formulas assume that the categories represent the tumor population of interest at the time being modeled, that each cell is assigned to one category, and that the positivity thresholds are defined consistently. They describe antigen-based reach, not proof of binding, synapse formation, or cell death. Functional response requires a separate calculation.

Why Marginal Antigen Prevalences Are Not Enough

Knowing that a certain proportion of cells are A-positive and a certain proportion are B-positive does not reveal how much those groups overlap. Because overlap determines the size of the union, marginal prevalences alone cannot determine dual-target coverage. This is especially important when data are reported separately for each marker rather than jointly at the single-cell level.

From Coverage to Predicted Killing

Let \(k_{ij}\) be the probability that a cell in subpopulation \(ij\) is killed over a specified treatment interval and under specified exposure conditions. The \(k_{ij}\) values may be estimated from suitable functional data or supplied by a mechanistic model. They need not be the same across antigen-positive groups: antigen density, accessibility, signaling, and other cell properties may differ. The total predicted fraction killed is the mixture-weighted average:

This result assumes the \(f_{ij}\) values describe the starting population for the interval, the \(k_{ij}\) values apply to that interval and exposure, and each starting cell belongs to exactly one subpopulation. It does not require killing outcomes in different cells to be independent; it uses their expected proportions. If antigen status or exposure changes materially during treatment, a time-varying or state-transition model is needed instead of applying one fixed mixture for the whole course.

A useful illustrative assumption is that each antigen-specific functional route kills a positive cell with a specified probability and that the two routes act independently on a dual-positive cell. If their probabilities are \(e_A\) and \(e_B\), the probability that neither route kills is \((1-e_A)(1-e_B)\). Therefore, the dual-positive killing probability under this assumption is \(k_{11}=1-(1-e_A)(1-e_B)\). This is a modeling choice, not a general property of bispecific agents: shared signaling limits, correlated responses, or other interactions can make the independence assumption inappropriate.

A Short Justification: Why the Mixture Average Applies

Imagine \(N\) starting cells, with approximately \(Nf_{ij}\) cells in category \(ij\). If a fraction \(k_{ij}\) of that category is killed, the expected number killed from the category is \(Nf_{ij}k_{ij}\). Adding across the mutually exclusive categories gives the expected total killed:

Dividing by \(N\) gives \(K_{\mathrm{total}}=\sum_{i,j}f_{ij}k_{ij}\). The calculation is a weighted average because each category contributes in proportion to its starting share. It would be incorrect to average the four \(k_{ij}\) values equally unless all four subpopulations had equal fractions.

Model Choices, Diagnostics, and Limitations

A four-category model is transparent and often useful when measurements support clear positive and negative groups. It is not the only option. If antigen abundance varies continuously, replacing each group with a single binary status can discard important information. A continuous-density model represents each cell by its A and B antigen levels and averages a response function over their joint distribution. In notation, if \(g(a,b)\) is the joint density of antigen levels and \(k(a,b)\) is the predicted killing probability at those levels, then total killing is \(\int\int k(a,b)g(a,b)\,da\,db\). This approach requires data or justified assumptions about the joint distribution and response function; separate average antigen levels do not generally specify either one.

There is also a difference between an empirical subpopulation model and a mechanistic binding-state model. The empirical approach uses \(k_{ij}\) values directly and can be practical when functional data are available. A mechanistic model may represent binding, bridging, or signaling states and derive \(k_{ij}\) from those processes. The state-based and synapse-signaling concepts provide a basis for such extensions, but they do not remove the need to represent the mixture of tumor phenotypes. More detailed models introduce more parameters and need corresponding measurements for identification.

Useful diagnostics begin with bookkeeping. Check that all four fractions are nonnegative and sum to one, and that the marginals agree with the joint fractions. Compare model-predicted killing with observed responses separately for antigen-defined subgroups where possible, rather than validating only against an overall average. Large systematic errors in one subgroup can be hidden by a plausible total. Sensitivity analysis over uncertain overlap and killing probabilities is also informative: if the predicted benefit of dual targeting changes substantially across plausible values, the conclusion depends on information the current data do not determine.

Other warning signs include assigning all antigen-positive cells the same response despite substantial density variation, treating marginal antigen prevalences as independent without evidence, and applying a fixed mixture after treatment has selected among the subpopulations. Positivity thresholds also need care: assay noise or threshold changes can move cells between categories and alter calculated coverage. When measurements are noisy or the boundaries are biologically gradual, a continuous or probabilistic classification may be more appropriate.

The independence calculation for two functional routes is particularly easy to overinterpret. It assumes that, conditional on the cell being dual-positive and on the specified exposure, the two route-specific killing events combine as independent opportunities. If both routes depend on the same limiting downstream process, then \(1-(1-e_A)(1-e_B)\) may overpredict killing. In that case, use an interaction model or estimate \(k_{11}\) from data rather than imposing the independence formula.