Generalized separable parameter space techniques for fitting 1K-5K serial compartment models

Kinetic modeling is widely used to analyze dynamic imaging data, estimating kinetic parameters that quantify functional or physiologic processes in vivo. Typical kinetic models give rise to nonlinear solution equations in multiple dimensions, presenting a complex fitting environment. This work generalizes previously described separable nonlinear least-squares techniques for fitting serial compartment models with up to three tissue compartments and five rate parameters.

The approach maximally separates the linear and nonlinear aspects of the modeling equations, using a formulation modified from previous basis function methods to avoid a potential mathematical degeneracy. A fast and robust algorithm for solving the linear subproblem with full user-defined constraints is also presented. The generalized separable parameter space technique effectively reduces the dimensionality of the nonlinear fitting problem to one dimension for 2K-3K compartment models, and to two dimensions for 4K-5K models.

Exhaustive search fits, which guarantee identification of the true global minimum fit, required approximately 10 ms for 2K-3K and 1.1 s for 4K-5K models, respectively. The technique is also amenable to fast gradient-descent iterative fitting algorithms, where the reduced dimensionality offers improved convergence properties. The objective function for the separable parameter space nonlinear subproblem was characterized and found to be generally well-behaved with a well-defined global minimum. Separable parameter space fits with the Levenberg-Marquardt algorithm required fewer iterations than comparable fits for conventional model formulations, averaging 1 and 7 ms for 2K-3K and 4K-5K models, respectively. Sensitivity to initial conditions was likewise reduced.

The separable parameter space techniques described herein generalize previously described techniques to encompass 1K-5K compartment models, enable robust solution of the linear subproblem with full user-defined constraints, and are amenable to rapid and robust fitting using iterative gradient-descent type algorithms.