Hi,

using one (mixed) model including the data from all patients is somewhat tricky, since day zero for each person is unknown, thus we do not know "where to start" with our time dependent variable. Which is actually the forcus of your question.

I would set up a regression model for each person, fit this to the data (introducing the data as a numerical variable with day = 0 for the fist existing measurement). Using the parameters from the regression model, you subsequently can set y=0 and find the corresponding x (which is a negative value telling you how many days before the first measurement the production of the marker started).

E.g. using bootstrap you can even calculate confidence intervals for the real "day zero"