A general algorithm for optimal sampling schedule design in nuclear medicine imaging

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Abstract

Optimal sampling schedule (OSS) is of great interest in biomedical experiment design, as it can improve the physiological parameter estimation precision and significantly reduce the samples required. A number of well designed algorithms and software packages have been developed, which deal with the instantaneous measurements at discrete times. However, in nuclear medicine tracer kinetic studies, the imaging systems, such as positron emission tomography (PET) and single photon emission computed tomography (SPECT), take measurements (images) based on continuous accumulation over time intervals. In this case, the existing algorithms cannot be used to design OSS so as to reduce the image frame numbers. In this paper, a general OSS design algorithm for the accumulative measurement is proposed. The potential usefulness of the algorithm is demonstrated by its designing OSS in [18F] fluoro-2-deoxy-d-glucose (FDG) studies with PET to estimate the local cerebral metabolic rate of glucose. The robustness of parameter estimation using the OSS with respect to intra-subject and inter-subject parameter variations is also presented.

Introduction

Optimal sampling schedule (OSS) is of great interest in biomedical experiment design, as its use can improve the physiological parameter estimation precision and significantly reduce the number of samples required. DiStefano introduces the advantages of using the experiment design as a tool in dynamic studies of biomedical systems in which very limited data is available [1]. Since then, the OSS design has been investigated extensively to provide guidelines for experiments to optimally arrange the limited number of samples and to conduct cost effective studies. Several well designed algorithms which deal with the instantaneous measurements at discrete times, have been developed for finding the OSS design in model-based physiological and pharmacological studies, based on the Fisher information matrix criterion [2], [3], [4], [5]. A comprehensive review on experiment design criteria can be found in [26].

In nuclear medicine tracer kinetic studies, however, the imaging systems, such as positron emission tomography (PET) and single-photon emission computed tomography (SPECT), take images (image frames) by dividing the total scanning time frame into many contiguous intervals and recording the accumulated radioactive counts continuously over each interval into each image frame. Due to this intrinsic difference from instantaneous sampling, the existing algorithms cannot be used to design OSS in order to improve the estimation quality and reduce the number of image frames. Furthermore, it is impossible to obtain replicated measurements from individual scanning intervals [4], [5], [7]. Therefore, the OSS design has to be reformulated in this new circumstance. For convenience, we shall use PET as our reference imaging system in the following description.

In this paper, a general algorithm for finding the OSS design for the accumulative measurement (which is the measurement of average count rates in a subinterval) is proposed. Several key novel steps incorporating the characteristics of the accumulative measurement and handling the interval merging are presented. The minimum number of image frames is systematically found and properly arranged, based on the D-optimal design criterion, given an initial sampling schedule and a set of a priori estimated parameters. The potential usefulness of the algorithm is demonstrated by its application to the design of [18F]fluoro-2-deoxy-d-glucose (FDG) PET studies to estimate the local cerebral metabolic rate of glucose (LCMRGlc). In addition, intra-subject and inter-subject parameter variations are investigated to evaluate the robustness of parameter estimation using the OSS approach.

Section snippets

The system-experiment model

Consider a single-input single-output (SISO) dynamic system [1], [5] on the observation interval [t0, T] with t0 and T representing the start and end times, respectively, andẋ(t,p)=f[x(t,p),u(t);p]x(t0,p)=x0y(t,p)=g[x(t,p);p]h[x(t,p),u(t);p]≥0z(tk)=y(tk)+e(tk),k=1,2⋯,Ny(tk)=1Δtktk−1tky(t)dtwhere x is a n-dimensional state vector; t represents the time abscissa; u and y are scalar input and output functions; f and g are linear or non-linear functions which describe the structure of the system

The iterative procedure

According to the information matrix approach as described in the previous section, a D-optimal sampling protocol SO is the one at which the cost function Ψ( · ) takes its maximum. The optimization problem is generally non-linear and an optimal design is therefore, obtainable numerically. Fig. 1 presents the flowchart of the iterative procedure for maximizing the cost function and finding the optimal sampling schedule.

The relaxation procedure, which is similar to the one given in [4] in terms of

Case study

The developed algorithm was applied to find an OSS design for the FDG model, which is used to quantitatively calculate the LCMRGlc with PET [19]. The three-parameter FDG model is used in this case study. The use of optimal sampling schedules together with the four-parameter [19] and five-parameter [20] FDG models can be found in [21].

Discussion

A general algorithm for optimal sampling schedule design in the context where measurements are obtained based on accumulation, has been presented. This general algorithm can automatically find the minimum number of image frames and properly arrange them using the D-optimality criterion. The accumulative measurement characteristics is incorporated into the OSS design process by using the measurement variance structure given in Eq. (8), which takes both the length of the scanning interval and the

Conclusions

We have developed a general optimal sampling schedule design algorithm for accumulative measurements, which fills the gap that the existing algorithms and software packages do not cover. Based upon the merging operation introduced, the algorithm can automatically find the minimum number of image frames required for parameter estimation and properly arrange them. Computer simulations based on real experimental data from dynamic brain FDG-PET studies have demonstrated the usefulness of the

Acknowledgements

This research is supported by ARC and UGC grants.

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