Application of the maximum likelihood principle to separate exponential terms in T2 relaxation of nuclear magnetic resonance.
The method of maximum likelihood has been implemented for the estimation of multiple exponential components of T2 decay curves in spin echo NMR measurements on biologic tissues. Each Each component contributes an exponential term described by two parameters (initial amplitude and T2) to the T2 decay curve. The maximum likelihood method estimates the parameters and their standard errors for all terms simultaneously, avoiding the subjectivity inherent in methods such as graphical peeling. In the model used, it was assumed that water protons are compartmentalized and that the measured spin echo signals from the protons undergoing relaxation obey the Poisson distribution. A system of non-linear equations was derived and solved iteratively for the values of the exponential parameters which maximize the likelihood of obtaining the observed data under these assumptions. The approach was implemented for bi- and tri-exponential models on a MicroVAX II computer (Digital Equipment Corporation, Maynard, MA). Simulations of bi- and tri-exponential data, with and without system noise, were analyzed to assess the accuracy and reproducibility of the method. A subset of the simulations was repeated with non-linear least squares techniques and was compared to the results obtained with maximum likelihood. Rabbit muscle and gerbil brain samples were measured and analyzed with the maximum likelihood method. The simulations showed that within specific limits on relative sizes and relaxation rates of components, these parameters can be estimated with errors less than 5%. The comparison to non-linear least squares analysis showed that the maximum likelihood method is generally superior in estimating the parameters in difficult cases. The results from tissue measurements demonstrate that the method is effective even in cases where graphical peeling would clearly not yield reliable results.
Sandor
T
T
Department of Radiology, Harvard Medical School, Boston, MA 02115.
Bleier
A R
AR
Ruenzel
P W
PW
Adams
D F
DF
Jolesz
F A
FA
D000465:Algorithms; D000818:Animals; D001921:Brain / Q000033:anatomy & histology; D005849:Gerbillinae; D006801:Humans; D008279:Magnetic Resonance Imaging / Q000379:methods*; D008961:Models, Structural; D009132:Muscles / Q000033:anatomy & histology; D011817:Rabbits