Unknown Dosing Histories

I'm in the process of designing a population PK design for a phase II efficacy study. Patients will be instructed to take their study medication three times a day, 1-2 hours before meals. Since this is an out-patient study, complete dosing histories will not be known. However, during each visit to the clinic for PK blood sampling, the patient will be asked to recall the times of their last 3 doses. Here's my question: What is the best way to code the model when the complete dosing history is not known? Some possible options are: i) assume equally-spaced dosing, or ii) assume a standard unequally-spaced dosing regimen perhaps indexed to typical meal times (e.g., 7am, 12noon, 6pm) since patients are instructed to take their study medication 1-2 hours before meals, or iii) for each patient, assume a daily dosing pattern to be fixed between visits based on the times of their last 3 doses prior to their visit to the clinic.

Re incomplete dosing records. You;ve taken a real tiger by the tail in this one. First off, there is no good approach. Period. If you can assume that the information is missing at random (that is, one is not missing data on those indivdiuals who tend to have kinetics different in some way than the rest), and you have a reasonable way to guess what the missing data might look like (that is, you know what distributin it is drawn from - say a uniform distribution with bounds +/- an hour from nominal dose time), then you could use a method called multiple imputation. This method essentially says to make up a bunch of complete data sets based on randomly generating the missing data (if the distribution of missing data depends on the unknown model parameters, then you have to use something called the EM algorithm - I actually don't know what you do when both are true - probably EM within each imputation). Then you analyse each imputed data set according to a fixed model (i.e., same set of covariates entering in same ways so the parameters are the same number and meaning) and use the results of all these analyses to compute a final set of estimates with appropriate standard errors (the latter are the average of the values you get from each fit, plus the std dev of the different estimates from the different fits). You can see that this is a formidable task using NONMEM. Not to mention the effect of incorrect data on the residual error model. The problem with just making up the data as you propose is that your standard errors will be too good as you are acting as though stuff you don't know was known. A former fellow of mine, Mats Carlsson, has come up wtih some things that help the residual error model in the case of errors in the data, and you can contact him (mats@c2355.ucsf.edu). You can find refrences to multiple imputation by looking for papers by Don Rubin in recent statistics literature. The above is not meant to solve your problem, but to let you know that there are no easy solutions .... Hopefully, you can examine your data and likely model and discover that the missing doses (those more than 3 doses ago) have very little effect on predictions (look at the partial derivative of the predictions wrt the missing dose(s)) - if so, then you have a non-problem and can do anything that seems reasonable. Hope I've been of some help,

Another possibility for modeling unknown dosing intervals is to measure the trough level at the days that the patients undergo blood sampling for PK analyses, and use this trough level as starting point for the current dose. As an example, assume a drug that behaves according to a one-compartment model, and is given orally three times a day for 30 days. Then, the morning plasma concentration-time curve at any day that the patients are being sampled can be described as follows: C = C0*EXP(-Ke*T) + (D/V)*(Ka*(Ka-Ke))*(EXP(-Ke*T)-EXP(-Ka*T)) where C0 is the trough level measured before the morning dose at a particular sampling day. This equation assumes that the absorption phase from the previous dose died out by the time that the next dose is administered. An approach that I have used successfully is to consider C0 as another parameter to be estimated. This is helpful if you do not have a trough level measurement, or if you want to allow some flexibility by incorporating error on the through level, which is usually the case since that level may be low and difficult to measure. Besides, you can get estimates of population mean and variability on C0. Another advantage of this approach is that it allows for diurnal variations. In the case that I used this, the morning trough levels were higher in the morning than during the day (assuming equally spaced dosing intervals). However, since blood sampling during the night was difficult, we couldn't really find out what was going on during the night (change in CL or Ka?), and implement this in the model. By making C0 a fixed effect, we eliminated that problem.

I really encourage you to consider a patient diary where doses can be logged. This will be far better than "making up the data" (see Lewis' comments below), even if there is some error in the diary entries.

I would like to 'support' Rene Braekman's suggestion of including an 'unknown' C0 in the model and getting a pre-study day dose blood sample. I've used this approach with more traditional PK modeling (ie non-NONMEM :-)) and it works. It doesn't even have to be at steady state - just post absorption (and post-distribution if two compartment). If two compartment - post-distribution you can derive Ct(0) [zero time tissue compartment concentration from Cp(0) and k12, k21] so it isn't another adjustable parameter.

I have often thought that an elegant study would be to use the Medication Events Monitoing System (MEMS, Aprex Corporation) to get real dosing histories for an outpatient study. I did a pilot study of this sort while a fellow and am planning a larger study of adolescent compliance/pk/response to HIV therapy. The MEMS should give reasonable data on when the doses are taken, and if the study is still in the design phase could be used.