GLLS for optimally sampled continuous dynamic system modeling: theory and algorithm

The original generalized linear least squares (GLLS) algorithm was developed for non-uniformly sampled biomedical system parameter estimation using finely sampled instantaneous measurements (D. Feng, S.C. Huang, Z. Wang, D. Ho, An unbiased parametric imaging algorithm for non-uniformly sampled biomedical system parameter estimation, IEEE Trans. Med. Imag. 15 (1996) 512-518). This algorithm is particularly useful for image-wide generation of parametric images with positron emission tomography (PET), as it is computationally efficient and statistically reliable (D. Feng, D. Ho, Chen, K., L.C. Wu, J.K. Wang, R.S. Liu, S.H. Yeh, An evaluation of the algorithms for determining local cerebral metabolic rates of glucose using positron emission tomography dynamic data, IEEE Trans. Med. Imag. 14 (1995) 697-710). However, when dynamic PET image data are sampled according to the optimal image sampling schedule (OISS) to reduce memory and storage space (X. Li, D. Feng, K. Chen, Optimal image sampling schedule: A new effective way to reduce dynamic image storage space and functional image processing time, IEEE Trans. Med. Imag. 15 (1996) 710-718), only a few temporal image frames are recorded (e.g. only four images are recorded for the four parameter fluoro-deoxy-glucose (FDG) model). These image frames are recorded in terms of accumulated radio-activity counts and as a result, the direct application of GLLS is not reliable as instantaneous measurement samples can no longer be approximated by averaging of accumulated measurements over the sampling intervals. In this paper, we extend GLLS to OISS-GLLS which deals with the fewer accumulated measurement samples obtained from OISS dynamic systems. The theory and algorithm of this new technique are formulated and studied extensively. To investigate statistical reliability and computational efficiency of OISS-GLLS, a simulation study using dynamic PET data was performed. OISS-GLLS using 4-measurement samples was compared to the non-linear least squares (NLS) method using 22-measurement samples, GLLS using 22-measurement samples and OISS-NLS using 4-measurement samples. Results demonstrated that OISS-GLLS was able to achieve parameter estimates of equivalent accuracy and reliability in comparison to NLS or GLLS using finely sampled measurements (22-measurement samples), or OISS-NLS using optimally sampled measurements (4-measurement samples). Further more, as fewer measurement samples are used in OISS-GLLS, this algorithm is computationally faster than NLS or GLLS. Therefore, OISS-GLLS is well-suited for image-wide parameter estimation when PET image data are recorded according to the optimal image sampling schedule.