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pd-and-exposure-response.md

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1# PD and exposure-response workflow

PD and exposure-response workflow

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3## Choose a structure the observations can distinguish

5| Structure | Equation/role | Main identification concern |

6| --- | --- | --- |

7| Direct Emax | `E0 + Emax*C/(EC50+C)` | Without a plateau, Emax and EC50 trade off |

8| Sigmoid Emax | `E0 + Emax*C^h/(EC50^h+C^h)` | Hill, EC50 and Emax need concentration-range support |

9| Effect compartment | `dCe/dt=ke0*(C-Ce)` | Delay can also arise from sampling, metabolites or turnover |

10| Indirect response | Production/loss modulated by drug | Baseline turnover and drug effect can be confounded |

11| Logistic | `logit(p)=a+b*exposure` | Separation, confounding and repeated observations |

13Effect-compartment Ce is a link variable, not an additional drug mass. Indirect response models I–IV

14inhibit/stimulate production or loss. An inhibitory fraction must stay between zero and one;

15stimulation terms must not make rates negative. Estimate/fix turnover with independent evidence

16where possible. Similar hysteresis loops do not uniquely select one mechanism.

18The ODE and regression helpers check finite inputs and solver/optimizer outcomes, but local

19standard errors are not evidence of structural identifiability. A converged fit with a deficient

20Jacobian must not imply precise parameters. Check profiles, bounds, starting-value sensitivity and

21predictions under perturbation, especially when the sampled range never reaches a plateau.

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.

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23## Exposure-response is not automatically causal

25Randomized dose does not randomize realized exposure. Clearance can covary with disease severity,

26body size, adherence and prognosis. Derived exposure carries estimation error; time-varying

27clearance and survival-dependent sampling can induce bias. Define the estimand, temporal ordering,

28confounders, missingness and sensitivity analyses before interpreting a slope as a dose benefit.

29Balance efficacy and safety outcomes rather than maximizing a single modeled response.

30[FDA exposure-response guidance](https://www.fda.gov/regulatory-information/search-fda-guidance-documents/exposure-response-relationships-study-design-data-analysis-and-regulatory-applications).

32`exposure_response.py` uses independent observations. Repeated longitudinal responses, count or

33time-to-event endpoints need models with appropriate distributions and dependence. Logistic data

34must contain both outcomes and exposure variation; complete/quasi-complete separation is rejected

35rather than returning a misleading finite estimate. C-QTc limitations are in [DDI/QT](ddi-and-qt.md).

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.

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37## Pharmpy model construction

39These transformations were executed independently on the bundled example model under Pharmpy 2.2.0.

40They return new models; they do not estimate parameters or establish adequacy:

42```python

43import pharmpy.modeling as m

44base = m.load_example_model("pheno")

45direct = m.set_direct_effect(base, expr="sigmoid")

46link = m.add_effect_compartment(base, expr="emax")

47turnover = m.add_indirect_effect(base, expr="emax", prod=True)

48```

50`prod=True` targets production; `False` targets elimination. Inspect the resulting symbolic model,

51observation mapping, parameter signs and baseline conditions before fitting it to actual data.

52[Pharmpy modeling API](https://pharmpy.github.io/latest/api/pharmpy.modeling.add_indirect_effect.html).

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.

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