Generalized Linear Least Squares Research Articles

Bound Generalized Linear Least Square (B-GLLS) Algorithm for Parametric Imaging with Pet

The recently developed generalized linear least squares (GLLS) fast algorithm is useful in image-wide parameter estimation to generate parametric images with positron emission tomography (PET). However, parameter estimation using the GLLS may not converge for every pixel in noisy clinical PET data. In this study, a bound generalized linear least squares (B-GLLS) algorithm was proposed to guarantee that the estimation converges for noisy pixel curves and the parameters estimated are within the physiological and pathological ranges. The clinical results demonstrated that the B-GLLS algorithm can generate more accurate parametric images and is potentially useful in dynamic clinical PET studies.

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). Furthermore, 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.

Rapid algorithms for the construction of cerebral blood flow and oxygen utilization images with oxygen-15 and dynamic positron emission tomography

Two rapid estimation algorithms for construction of cerebral blood flow (CBF) and oxygen utilization (CMRO) images with dynamic positron emission tomography (PET) are presented. These algorithms are based on the linear least squares (LLS) and generalized linear least squares (GLLS) methodologies. Using the conventional two-compartmental model and multiple tracer studies, we derived a linear relationship for brain tissue activity to arterial blood activity, time-integrated arterial blood activity and time-integrated brain tissue activity. The LLS technique is computationally efficient as no regression analysis is required, while GLLS is used to refine the estimates obtained from LLS. A comparative study using non-linear least squares regression (NLS) revealed excellent correlation between the new algorithms for various noise levels expected in clinical applications. A sensitivity analysis was performed to examine reliability and identifiability of the parameter estimates. In view of the results, LLS and GLLS provide rapid and reliable estimates of CBF and CMRO when applied to dynamic PET data. These algorithms are particularly suitable for pixel-by-pixel construction of high resolution and highly accurate PET functional images.

Generalized linear least squares algorithm for non-uniformly sampled biomedical system identification with possible repeated eigenvalues

The recently developed generalized linear least squares (GLLS) algorithm has been found very useful in non-uniformly sampled biomedical signal processing and parameter estimation. However, the current version of the algorithm cannot deal with signals and systems containing repeated eigenvalues. In this paper, we extend the algorithm, so that it can be used for non-uniformly sampled signals and systems with/without repeated eigenvalues. The related theory and detailed derivation of the algorithm are given. A case study is presented, which demonstrates that the extended algorithm can provide more choices for system identification and is able to select the most suitable model for the system from the non-uniformly sampled noisy signal.

An unbiased parametric imaging algorithm for nonuniformly sampled biomedical system parameter estimation

An unbiased algorithm of generalized linear least squares (GLLS) for parameter estimation of nonuniformly sampled biomedical systems is proposed. The basic theory and detailed derivation of the algorithm are given. This algorithm removes the initial values required and computational burden of nonlinear least regression and achieves a comparable estimation quality in terms of the estimates' bias and standard deviation. Therefore, this algorithm is particular useful in image-wide (pixel-by-pixel based) parameter estimation, e.g., to generate parametric images from tracer dynamic studies with positron emission tomography. An example is presented to demonstrate the performance of this new technique. This algorithm is also generally applicable to other continuous system parameter estimation.

An Efficient Estimation Method for Biomedical System Modeling

A generalized linear least squares (GLLS) algorithm for non-uniformly sampled biomedical system parameter estimation is proposed. This algorithm is generally applicable to a variety of models with different structures and can produce unbiased parameter estimation. As this algorithm requires very little computing time, it can potentially be used for generating parametric images.