Worked ADC Model of Bystander Killing Across DAR

From DAR Species to Bystander Killing

An ADC with a heterogeneous drug-to-antibody ratio (DAR) is a mixture of species, not a single molecule with the mean DAR. Those species can differ in payload delivery and intracellular release, and released payload may leave an ADC-positive source cell, enter the local extracellular space, and reach neighboring cells. A worked model makes these steps explicit and lets us compare predicted killing across DAR distributions.

The model below focuses on a source-cell population and a neighboring-cell population. It represents payload exposure with a small set of compartments and translates integrated exposure into killing. Tumor and cellular complexities are condensed into model parameters; the resulting predictions are conditional on those assumptions, not direct clinical forecasts.

The factor \(\eta_d\) is a model input, not automatically equal across DAR species. It may summarize differences in delivery or uptake represented outside this reduced model. Payload release from \(U_d\) occurs at rate \(k_{\mathrm{rel},d}\); unreleased payload is also lost at rate \(k_{\mathrm{deg},U}\). Active source-cell payload is cleared at rate \(k_{\mathrm{clear},H}\) and exits to the extracellular space at rate \(k_m\). Extracellular payload is cleared at rate \(k_{\mathrm{clear},E}\) or taken up by neighboring cells at rate \(k_u\). Neighboring-cell payload is cleared at rate \(k_{\mathrm{clear},L}\).

All states are zero at time zero except \(U_d(0)=D f_d d\eta_d\). Rate constants are positive and have units of inverse time. The model assumes first-order release, loss, diffusion, and uptake; constant rates over the modeled interval; and additive contributions from DAR species. It treats the source and neighboring cells as well-mixed populations, with no spatial gradients or feedback from cell killing onto payload kinetics. These assumptions allow integrated exposures to be calculated directly.

Deriving Integrated Payload Exposure

For any state \(Z(t)\), define its area under the concentration-or-amount curve as \(\mathrm{AUC}_Z=\int_0^\infty Z(t)\,dt\). Here the units are normalized payload exposure units·h. Integrating each differential equation from zero to infinity uses the condition that the states eventually return to zero. For example, integrating the \(H_d\) equation gives \(H_d(\infty)-H_d(0)=k_{\mathrm{rel},d}\mathrm{AUC}_{U,d}-(k_{\mathrm{clear},H}+k_m)\mathrm{AUC}_{H,d}\). Since both boundary values of \(H_d\) are zero, the two integrated fluxes are equal.

Here \(Q_{\mathrm{rel},d}\) is the total amount released from the unreleased pool. Total AUC in each compartment is the sum across species: \(\mathrm{AUC}_H=\sum_d\mathrm{AUC}_{H,d}\), with corresponding sums for \(E\) and \(L\). The relations follow from integrated mass balance, so they do not require the full time courses to be solved. They do require positive loss rates so that the relevant exposures are finite.

Keep the Different Bystander Fractions Separate

The equations imply several useful ratios, but they describe different quantities. The fraction of source-cell payload eliminated by diffusion is a fraction of the total loss from \(H\). The ratio of extracellular to source-cell AUC instead compares exposures in two compartments. Finally, the fraction of extracellular payload taken up by neighboring cells is a fraction of extracellular loss.

For the worked parameter values, the source-cell diffusion-loss fraction is \(0.20/(0.30+0.20)=0.40\), the ratio \(\mathrm{AUC}_E/\mathrm{AUC}_H\) is \(0.20/(0.20+0.20)=0.50\), and the fraction of extracellular payload taken up is also \(0.20/(0.20+0.20)=0.50\). The last two happen to be equal for these rates; they are not interchangeable definitions. In general, \(\mathrm{AUC}_E/\mathrm{AUC}_H\) depends on extracellular clearance and uptake, not on source-cell clearance.

Translating Exposure into Cell Killing

For a simple exposure-response link, assume each cell’s cumulative killing hazard is proportional to its active payload AUC. Let \(\beta_H\) and \(\beta_L\) be potency coefficients for source and neighboring cells, respectively, in inverse normalized exposure units·h. The survival fraction is the exponential of the negative cumulative hazard, and the killed fraction is one minus survival. This is a specified model assumption, not a consequence of the payload kinetics.

If the modeled tumor population contains fraction \(w_H\) of source cells and fraction \(w_L=1-w_H\) of neighboring cells, its predicted killed fraction is \(K_{\mathrm{tumor}}=w_HK_H+w_LK_L\). This weighting assumes the two cell groups are exhaustive and that their fractions are specified independently of the killing calculations.

Assumptions, Diagnostics, and Alternatives

Several checks help identify when this reduced model is being used beyond its support. First, check units: every rate must have units of inverse time, each AUC has exposure-units·time, and each \(\beta\mathrm{AUC}\) in the exponential must be dimensionless. Second, verify that DAR fractions are nonnegative and sum to one, and that every integrated input and AUC is nonnegative. Third, check mass balance at each compartment: for example, \(Q_{\mathrm{rel},d}=(k_{\mathrm{clear},H}+k_m)\mathrm{AUC}_{H,d}\). If a numerical solver produces a negative state or violates this equality materially, inspect the equations, tolerances, and units.

The model also gives interpretable limiting checks. If \(k_m=0\), then \(\mathrm{AUC}_E=\mathrm{AUC}_L=0\), so this pathway predicts no bystander killing. If \(k_u=0\), extracellular payload can be present but neighboring-cell exposure through uptake is zero. If \(k_{\mathrm{clear},L}\) is very large, neighboring-cell AUC becomes small. These checks can reveal coding errors and clarify which process drives a prediction.

A frequent mistake is to treat the mean DAR as though every ADC molecule carried that DAR and had the same release behavior. That shortcut is justified only when species differences do not affect delivery, release, or any other modeled process, or when the approximation is shown to preserve the relevant outputs. Here, DAR 2 and DAR 4 have different release rates and delivery factors, so averaging DAR first would discard information needed to calculate \(Q_{\mathrm{rel}}\). Another common error is to call \(\mathrm{AUC}_E/\mathrm{AUC}_H\) the fraction of source payload diffusing. It is an exposure ratio governed by extracellular loss, whereas the source diffusion-loss fraction is \(k_m/(k_{\mathrm{clear},H}+k_m)\).

This compartmental model is simpler than a spatial reaction-diffusion model, which can represent distance-dependent gradients, or a cell-level model that tracks heterogeneous uptake and susceptibility. It is more mechanistic than a direct killing model based only on administered dose, because it represents release and transfer explicitly. Its limitations include first-order kinetics, fixed cell populations, no spatial structure, and a simple exponential killing hazard. If payload exposure is highly localized, rates vary over time, or cells differ substantially in sensitivity, these assumptions may be inadequate. The model is best used to organize hypotheses and compare scenarios with transparent assumptions, not to claim that a particular DAR distribution must be superior in every tumor.