References for the B-cell indirect-response estimation document
The theory of Emax identifiability, and the published B-cell depletion models
Nothing here has been read in full. Entries 5 to 8 were extracted from the full papers by a fetch tool on 2026-09-09 and carry ⚠️: the numbers came from the paper, and the transcription has not been checked by eye. Every other entry is from an abstract or a search summary and carries ❌. The simulations in the working document do not depend on any of them; the parameter values were chosen to resemble entries 5 to 7 and the document says so where it uses them.
The working document is the EC50 identifiability page, and the folder’s other documents are on the project index.
Status markers
- ❌ Not checked. Characterization from an abstract or a search result.
- ⚠️ Transcribed, unverified. A number or model structure in the working document came from the full source through a fetch tool, and has not been checked against it by eye.
- ✅ Checked. Verified against the source. None yet.
Where to start
- Entry 1 first. It is the working document’s Section 8 in its original form, twenty-eight years earlier: when the data do not reach \(E_{\max}\), report \(E_{\max}/EC_{50}\). Half an hour.
- Entry 2 next, for the simulation evidence that a fit can be good while \(E_{\max}\) and \(EC_{50}\) are both wrong, which is Section 6.
- Entries 5 and 7 together, as the two ends of the published record: the design that identified \(EC_{50}\) and the design that gave it up. An hour for both, on the parameter tables and the design descriptions.
- Entry 4 for the indirect-response model itself, if the four Jusko types are not already familiar.
Theory
1. Schoemaker, van Gerven and Cohen 1998 — estimating potency without attaining maximal effects ❌
J Pharmacokinet Biopharm 1998;26:581-593. doi:10.1023/A:1023277201179 · PubMed 10205772
What it says, from the abstract: \(E_{\max}\)-model parameters can be estimated with reasonable precision only if the curve shows signs of reaching a maximum; otherwise both \(EC_{50}\) and \(E_{\max}\) may be extremely imprecise. Proposes the parameter \(S_0 = E_{\max}/EC_{50}\), which characterizes potency adequately without a maximum.
The question. Whether the paper treats the case where the maximum is reached but is censored, which is the B-cell case, or only the case where it is never approached. The conclusion is the same; the argument may differ.
2. Dutta, Matsumoto and Ebling 1996 — sigmoid Emax parameters from truncated clinical data ❌
J Pharm Sci 1996;85:232-239. doi:10.1021/js950067y · PubMed 8683454
What it says, from the abstract: when the highest measured effect was below 95% of \(E_{\max}\), \(E_{\max}\) and \(EC_{50}\) were poorly estimated, with high coefficients of variation and pronounced bias, while the fit to the data was good and descriptors within the data range were estimated precisely.
The question. The size of the simulation and the bias direction, to compare with Section 6’s ridge.
3. Dayneka, Garg and Jusko 1993 — four basic indirect response models ❌
J Pharmacokinet Biopharm 1993;21:457-478.
The origin of the model in Section 1. Stimulation of \(k_{\rm out}\) is the fourth of the four. Not yet retrieved.
Published B-cell depletion models
The table in Section 7 of the working document summarizes these; the entries below carry the numbers and the questions.
5. Yu et al. 2022 — ofatumumab population PK-B cell model in relapsing MS ⚠️
CNS Drugs 2022. doi:10.1007/s40263-021-00895-w · PMC8927028
Transcribed. Quasi-steady-state TMDD for PK; indirect response on B cells with sigmoid \(E_{\max}\) stimulation of lysis by total ligand concentration. Five studies, 1,486 patients: 100 to 700 mg intravenous; the MIRROR Phase 2 at 3, 30 and 60 mg every 12 weeks and 60 mg every 4 weeks subcutaneous; the 20 mg monthly Phase 3 regimen. \(EC_{50}\) 0.0057 mg/L (RSE 8.5%), \(E_{\max}\) 159 (RSE 3.2%), \(k_{\rm out}\) 0.0124 per day (RSE 5.2%), baseline 194 cells/µL. \(K_D\) and \(k_{\rm off}\) fixed to preclinical values. “The 10-mg dose did not appear to achieve a similar level of depletion as the 20-mg dose until after 6 months.”
The question. The B-cell sampling schedule in MIRROR during the first week, and the depletion profiles of the 3 mg arm: whether they sit in the band of Section 3, and whether the fall was sampled.
6. Rituximab CD19+ suppression, population PD in paediatric patients ⚠️
Pharmaceutics 2023;15(11):2534. doi:10.3390/pharmaceutics15112534 · PMC10674351
Transcribed. Turnover model on CD19+ with rituximab amount raising the degradation rate through \(E_{\max} A / (ED_{50} + A)\); a linear form fit significantly worse, a sigmoid no better. 63 patients, 281 measurements, 52% below the limit. \(E_{\max}\) 155 (RSE 23%, CI 153 to 713), \(ED_{50}\) 0.692 mg (RSE 61%, CI 0.177 to 2.259), \(k_{\rm deg}\) 0.004 per day, CD19 half-life 173 days, rituximab half-life 11.6 days. “The studied dose-range and the highly efficacious doses may have contributed to hampering a precise estimation.”
The question. Why the linear form fit worse if the ridge argument holds. The likely answer is the early samples: a linear effect at high amount gives an implausibly fast fall. That would be Section 4’s first region doing its work, and their sampling times would show it.
7. Yan et al. 2022 — inebilizumab PD modelling in NMOSD ⚠️
Br J Clin Pharmacol 2022. doi:10.1111/bcp.15332 · PMC9545531
Transcribed. Haematopoietic transit model with inebilizumab accelerating removal from every compartment. 213 subjects across 0.1 to 10 mg/kg and 30 to 600 mg. “The estimated EC50 … was less than the assay lower limit of quantitation due to the high potency of inebilizumab,” so the effect became \(k = \text{slope} \cdot \log C\): slope 0.0140 (SE 0.0006), lifespan 391 days (SE 50), baseline 135 cells/µL; between-patient CV 52% on slope and 126% on lifespan; proportional residual error 61%.
The question. Whether the Phase 1 dose-ranging arms at 0.1 and 0.3 mg/kg produced partial depletion, and if so why they did not carry \(EC_{50}\). The working document’s answer is that they sat below the band; the paper’s profiles would confirm or refute it.
8. Rituximab population PK/PD in primary membranous nephropathy ⚠️
Transcribed. Quasi-steady-state TMDD with elimination of the drug-CD20 complex standing in for lysis; no separate \(E_{\max}\). 41 patients, 220 CD20+ counts, sampled before each infusion. \(k_{ss}\) 6.21 per µmol at 60% RSE.
9. Baltcheva et al. 2018 — model-based dose selection for ianalumab in Sjögren’s ❌
PAGE 2018. abstract
Two-compartment linear PK; indirect-effect B-cell turnover; a separate tissue receptor-occupancy model. No parameter estimates in the abstract.
10. Bar-Or et al. 2018 — the MIRROR study of subcutaneous ofatumumab ❌
Neurology 2018. PMC5957306
232 patients randomized to 3, 30 or 60 mg every 12 weeks, 60 mg every 4 weeks, or placebo; dose-dependent CD19 depletion. The data behind entry 5’s low-dose arms, and where the sampling schedule for the question under entry 5 would be found.