Abstract
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.
Highlights
Dynamic imaging techniques can measure and characterize temporal changes in imaging signals for many modalities, providing more indepth functional information than static images that provide a snapshot or time-average over the072502-1 Med
The archetypal example kinetic model is the compartment model,1–4 which comprises a series of homogenous compartments driven by an input function, and where temporal exchange between compartments is governed by rate parameters and simple linear differential equations
The separable parameter space techniques discussed in this paper generalize previously proposed methods for applying separable nonlinear least-squares to the compartment model fitting problem
Summary
Dynamic imaging techniques can measure and characterize temporal changes in imaging signals for many modalities, providing more indepth functional information than static images that provide a snapshot or time-average over the. This paper is focused on the mathematical problem of fitting compartment model equations to dynamic imaging data in order to estimate best-fit kinetic rate parameters for a predetermined fitting criterion (e.g., weighted least-squares). Other related issues, such as measurement of the input function or selection of fitting weights, fall outside the scope of this work and are not discussed. The nonlinear components of the reformulated equations have largely been interpreted as forming basis functions for the kinetic model, and the fitting algorithm searches among these bases to identify (weighted sums of) curves that best-match the measured data. The performance of both exhaustive search and the iterative LevenbergMarquardt algorithms are studied for separable parameter space fits and compared to fits using the conventional model formulations
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