Reduced parameter space formulations for fast and robust kinetic modeling

Objectives Dynamic imaging with compartment modeling can quantify in vivo physiologic processes. However, such models require solving multi-dimensional nonlinear optimization problems. Non-negative least squares (NNLS) provides perhaps the most robust solution, but can be time consuming and sensitive to initial conditions. Accelerated approaches are widely used, but may be less robust for noisy data. We propose theoretical reformulations that reduce the 2-5 dimensional nonlinear fitting space to only 1 (when k4=0) or 2 unknowns. These formulations greatly simplify the fits, make no assumptions beyond the weighted least-squares criterion, and do not require transformations that introduce noise correlations.

Methods The reduced parameter space formulations reorganize the compartment model solution equations into a sum of temporal terms with only 1 (k4=0) or 2 unknowns. All other parameters are eliminated by constraining the solution space to only allow solutions that are least-squares in the linear sense. The result is a nonlinear fitting problem in 1 or 2 unknowns. The constraint can also be applied in a “one-step late” (OSL) sense to simplify computations. The new formulations were implemented with several minimization algorithms, including exhaustive search and gradient methods, and tested using populations of simulated time-activity curves.

Results The reformulated objective functions are well-behaved and easily fit, providing fits identical to fully-converged conventional NNLS. For a 3-compartment model with vascular term and K1-k3, Levenberg-Marquardt consistently reaches global minimum in under 10 iter (single thread CPU time < 0.001s). Exhaustive search of the entire solution space to precision 0.001 takes < 0.02s. No local minima or convergence issues have been observed.

Conclusions Reduced parameter space formulations enable extremely fast and robust compartment modeling, and can be used to guarantee the global minimum to a given precision via exhaustive search. In addition, ultrafast implementation with gradient-descent algorithms enables full voxelwise parametric fitting in under 1 minute