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

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FIG. 1.

Generic serial compartment models, each consisting of an input driving 1–3 tissue compartments in series that exchange according to the labeled rate parameters. We use a shorthand nomenclature to quickly reference each generic model as shown. For example, the “3K” model refers to the model with input plus two additional compartments and three rate parameters.

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FIG. 2.

Example time-activity curves for 2K-4K serial compartment models simulated using the conventional and separable parameter space model formulations. Three example curves are shown for each model. The curves for the conventional and reformulated models were identical up to the numerical precision of the computer, and overlap exactly in the plots.

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FIG. 3.

Plots of the separable parameter space objective function for an example noisy time-activity curve with 3K compartment model. The left plot shows how the values of κ, κ, and κ that minimize change as a function of υ (recall υ = + for the 3K model). Similarly, the plot at right shows the same objective function with the corresponding values of , , , and at each point. The objective function is well behaved, having a single clearly defined minimum with no other local minima, shelves, or confounding structures.

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FIG. 4.

Example separable parameter space objective functions for fitting a 3K compartment model to curves at five different noise levels. The minimum is deeper and better-defined for lower noise data, and becomes shallower as noise increases. However, for all noise levels the reformulated objective function remained well-behaved, and no complex topological features appeared with increasing noise levels.

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FIG. 5.

Contour plots of two examples separable parameter space objective functions for 4K (left) and 5K (right) compartment models. The grayscale represents values, and contour lines are drawn at regular intervals. Recall that the separable parameter space formulation for 4K-5K models has two nonlinear unknowns (υ, υ), and also that υ is constrained to be less than υ (hence the plots are only defined for regions υ < υ). The objective functions are generally well behaved and do not show complex topological features. However, some complex structure exists in the lower-left corner near υ = υ line and near υ = 0 which could slow convergence of gradient-descent type algorithms in these regions. These structures arise from “attraction” due to imaginary poles in the excluded υ > υ and υ < 0 regions of the solution space.

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FIG. 6.

Scatter plots of the 3K model objective function, macroparameters, and individual rate parameters comparing conventional versus separable parameter space fit results for noisy populations of time-activity curves. In each case, results for three different numbers of iterations are shown for the conventional fits with simulated annealing, demonstrating convergence effects as progressively higher numbers of iterations produced successively better fits (lower ) for some curves. Note that in all cases the separable parameter space fits with exhaustive search produced equal or lower than the conventional model fits to within the numerical precision of the search (approximately 1 part in 10). As a population, the conventional model fits approached the separable parameter space fits as the number of iterations increased.

Image of FIG. 7.

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FIG. 7.

Scatter plots of the 4K model objective function, macroparameters, and individual rate parameters ( not shown) comparing conventional vs separable parameter space fit results for noisy populations of time-activity curves. The conventional model results are shown for 10, 10, and 10 iterations of simulated annealing, showing convergence effects similar to those shown in Fig. 6 for the 3K model (but at 10× more iterations due to the increased complexity of the 4K model). Again, the separable parameter space exhaustive search fits provided the lowest for all cases to within the precision of the search, and the conventional model fits progressive convergence toward these global minima across the full population of curves with increasing iteration.

Tables

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TABLE I.

Nonlinear parameter definitions for 1K-5K compartment models.

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TABLE II.

Linear parameter definitions for 1K-5K compartment models.

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TABLE III.

Recovery of kinetic rate parameters.

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TABLE IV.

Levenberg-Marquardt fit results.

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TABLE V.

CPU times for fits.

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/content/aapm/journal/medphys/40/7/10.1118/1.4810937
2013-06-24
2014-04-24

Abstract

Kinetic modeling is widely used to analyze dynamic imaging data, estimating kinetic parameters that quantify functional or physiologic processes. 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.

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Scitation: Generalized separable parameter space techniques for fitting 1K-5K serial compartment models
http://aip.metastore.ingenta.com/content/aapm/journal/medphys/40/7/10.1118/1.4810937
10.1118/1.4810937
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