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获取原文件references/pd-and-exposure-response.md
Choose a structure the observations can distinguish
| Structure | Equation/role | Main identification concern |
|---|---|---|
| Direct Emax | E0 + Emax*C/(EC50+C) |
Without a plateau, Emax and EC50 trade off |
| Sigmoid Emax | E0 + Emax*C^h/(EC50^h+C^h) |
Hill, EC50 and Emax need concentration-range support |
| Effect compartment | dCe/dt=ke0*(C-Ce) |
Delay can also arise from sampling, metabolites or turnover |
| Indirect response | Production/loss modulated by drug | Baseline turnover and drug effect can be confounded |
| Logistic | logit(p)=a+b*exposure |
Separation, confounding and repeated observations |
Effect-compartment Ce is a link variable, not an additional drug mass. Indirect response models I–IV inhibit/stimulate production or loss. An inhibitory fraction must stay between zero and one; stimulation terms must not make rates negative. Estimate/fix turnover with independent evidence where possible. Similar hysteresis loops do not uniquely select one mechanism.
The ODE and regression helpers check finite inputs and solver/optimizer outcomes, but local standard errors are not evidence of structural identifiability. A converged fit with a deficient Jacobian must not imply precise parameters. Check profiles, bounds, starting-value sensitivity and predictions under perturbation, especially when the sampled range never reaches a plateau.
Exposure-response is not automatically causal
Randomized dose does not randomize realized exposure. Clearance can covary with disease severity, body size, adherence and prognosis. Derived exposure carries estimation error; time-varying clearance and survival-dependent sampling can induce bias. Define the estimand, temporal ordering, confounders, missingness and sensitivity analyses before interpreting a slope as a dose benefit. Balance efficacy and safety outcomes rather than maximizing a single modeled response. FDA exposure-response guidance.
exposure_response.py uses independent observations. Repeated longitudinal responses, count or
time-to-event endpoints need models with appropriate distributions and dependence. Logistic data
must contain both outcomes and exposure variation; complete/quasi-complete separation is rejected
rather than returning a misleading finite estimate. C-QTc limitations are in DDI/QT.
Pharmpy model construction
These transformations were executed independently on the bundled example model under Pharmpy 2.2.0. They return new models; they do not estimate parameters or establish adequacy:
import pharmpy.modeling as m
base = m.load_example_model("pheno")
direct = m.set_direct_effect(base, expr="sigmoid")
link = m.add_effect_compartment(base, expr="emax")
turnover = m.add_indirect_effect(base, expr="emax", prod=True)
prod=True targets production; False targets elimination. Inspect the resulting symbolic model,
observation mapping, parameter signs and baseline conditions before fitting it to actual data.
Pharmpy modeling API.