I have a dataset of 134 lung cancer patients who had an inflammatory biomarker measured at 3 time points before during and after therapy - we're trying to investigate if the biomarker is predictive of their survival risk. Thing is, the time between the three measurements is different for each patient so we want to calculate the slope per day to account for the unequal time between the measurements - does that make sense and if so how to do so?
Likely more context is needed to provide insight into the proposed analysis, but the general approach can be laid out:
The task is to use time varying covariates to estimate the hazard ratio for death comparing groups differing by a "slope" of pharmacokinetic trend in the inflammatory marker.
The Cox-model is a regression model that estimates hazard ratios in failure analyses. When the covariates change over time, you censor the patient at the timepoint of change, and re-enter them into the analysis with the updated covariate value.
The actual representation of the slope for a patient who's probably receiving multiple treatments (and multiple PK assessments) over time leaves a lot of open ended questions. For instance, should the covariate be the last dose administered, or the highest expressed slope among previous treatments, or the cumulative slope over each treatment administered? Is the dose administered at a particular infusion confounded by prior performance or complications? The strategy and rationale for handling any (or all) of these is to be laid out by the clinical expert.