  We compare and contrast the transit compartment ordinary differential
equation modelling approach with distributed and discrete delay differential
equation models. We focus on Quartino's extension to the Friberg transit
compartment model of myelosuppression, widely relied upon in the pharmaceutical
sciences to predict the neutrophil response after chemotherapy, and on a QSP
delay differential equation model of granulopoiesis. We extend the Quartino
model by considering a general number of transit compartments and introduce an
extra parameter which allows us to decouple the maturation time from the
production rate of cells, and review the well established linear chain
technique from the delay differential equation (DDE) literature which can be
used to reformulate transit compartment models with constant transit rates as
distributed delay DDEs. We perform a state-dependent time rescaling of the
Quartino model in order to apply the linear chain technique and rewrite the
Quartino model as a distributed delay DDE, which yields a discrete delay DDE
model in a certain parameter limit. We then perform stability and bifurcation
analyses on the models to situate such studies in a mathematical pharmacology
context.
  We show that both the original Friberg and the Quartino extension model
incorrectly define the mean maturation time, essentially treating the
proliferative pool as an additional maturation compartment, which can have far
reaching consequences on the development of future models of myelosuppression
in PK/PD.
