  Purpose: To derive kinetic equations for multi-compartmental contrast agent
distribution from first principles and apply it to two and three compartments
for Gd-EOB-DTPA using low time resolution human liver DCE-MRI data.
  Methods: The continuity and diffusion equation were combined and used to
derive a general form for differential equations governing multi-compartmental
particle exchange. They were applied to two (equivalent to the Tofts model) and
three compartments. Both models were fit to human DCE-MRI data with low
temporal resolution and three compartment model's parameters' implications are
discussed.
  Results: The model derived for two compartments is shown to be equivalent
with the Tofts model. Using reasonable biological and physical assumptions an
analytical solution for the three compartment model is obtained. The three
compartment model was able to fit all Gd-EOB-DTPA DCE-MRI data whereas the
Tofts model did not. Differences were increased for cases with large as well as
rapid uptake of contrast.
  Conclusion: We demonstrated the ability to fit sparse Gd-EOB-DTPA DCE-MRI
data using a three compartment model derived from first principles whose
parameters can be used to help quantify overall and regional liver function.
