TY - JOUR
T1 - Closed-form formulas for estimation of kinetic parameters in one- and multi-compartment models
JF - Journal of Nuclear Medicine
JO - J Nucl Med
SP - 2030
LP - 2030
VL - 52
IS - supplement 1
AU - Zeng, Gengsheng
AU - Kadrmas, Dan
AU - Gullberg, Grant
Y1 - 2011/05/01
UR - http://jnm.snmjournals.org/content/52/supplement_1/2030.abstract
N2 - 2030 Objectives This research develops closed-form formulas for estimating kinetic parameters for single and multiple compartment models in dynamic tomographic imaging. Methods Linear system theory is used to derive the closed-form solutions. The main derivation steps are: (1) Express the first order differential equations and measurement equation in the form of state equations. (2) Use the Laplace transform to convert the state equations to the system transfer function, where the kinetic parameters are related to the coefficients in the transfer function. (3) Obtain an analytical expression for the input-output relation in the continuous time domain. (4) Obtain expressions for time-delayed outputs and relate them using a single, novel "relation equation". (5) Solve the relation equation by inverting a small square matrix of dimension equal to the number of rate parameters, and (6) recover estimates of each kinetic parameter. Results The closed-form formula derivation is generic, with solution for each kinetic parameter uniquely determined by the formulas. Computer simulations show that exact kinetic parameters are recovered for noiseless data. For noisy data, estimates of the data variance can be incorporated into the relation equation to model the noise, providing weighted least squares estimation. Conclusions Commonly used non-linear fitting methods for kinetic parameter estimation are time-consuming and can run into problems with non-unique-solutions, prompting interest in linearized estimation techniques. The proposed method provides an alternative formulation to generalized linear least squares (GLLS) that is not subject to bias from the noise correlations that arise in conventional linear least squares. It is less sensitive to numerical integration errors, does not require continuous time-activity curves, and has less stringent temporal sampling requirements. This promising new technique merits further evaluation for dynamic SPECT and PET applications
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