Purpose To map T1 and the local flip angle (B1+) in human brain using a single MP3RAGE sequence with 3 rapid acquisitions of gradient echoes (RAGEs). Theory and methods A third RAGE with a relatively high flip angle was appended to an MP2RAGE sequence. Through curve fitting and a rational approximation for small flip angles and short TR, closed form solutions for T1 and B1+ were derived. The influence of different k‐space encoding schemes on precision and whether edge enhancement artifacts could be reduced with a saturation pulse applied prior to the third RAGE were explored. Validation of T1 estimates was performed using single‐slice inversion recovery (IR) and a subsequent region‐of‐interest–based comparison, whereas validation of B1+ was performed using a whole brain pixelwise comparison to a DREAM flip angle mapping protocol. Lastly, MP3RAGE was compared to T1‐mapping by MP2RAGE with separate B1+ correction. Results Whole brain maps of T1 and B1+ at 1 mm isotropic resolution were obtained with MP3RAGE in 06:37 min. A linear–reverse centric–reverse centric phase‐encoding order of the 3 RAGEs improved precision, and artifacts were successfully reduced with the saturation pulse. Estimations of T1 and B1+ deviated +2.5 ± 3.1% and −1.7 ± 8.6% from their respective references. Conclusion T1 and B1+ can be mapped simultaneously using MP3RAGE. The approach can be thought of as combining MP2RAGE with a dual flip angle T1‐mapping protocol. Both maps can be solved for analytically and will be inherently co‐registered at the high resolution associated with MPRAGE.
In this paper, the branch and bound technique used to solve a single machine scheduling problem, which is the problem of scheduling n-job on a single machine of multi-objective function with triangle fuzzy due date numbers which are formulated as 1 \ D ˜ j = T F N \ ∑ j = 1 n C j + L max . The target of this paper is to obtained optimal sequence of our problem. The computational results are calculated by using Matlab program and compare the results with the complete enumeration method.
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