Magnetic Resonance in Medicine 63:1431–1436 (2010)
Concatenated and Parallel Optimization for the Estimation of T1 Map in FLASH MRI With Multiple Flip Angles Defeng Wang,1,2 Lin Shi,2* Yi-Xiang J. Wang,2 Jing Yuan,2 David K.W. Yeung,2 Ann D. King,2 Anil T. Ahuja,2 and Pheng Ann Heng1 Most traditional methods for T1 map estimation in MRI with fast low-angle-shot sequences are aimed at high efficiency by compromising the fitting accuracy. In this paper, the fundamental problem of parameter estimation in fast low-angle-shot MRI was re-examined, and an accurate and fast optimization approach, named concatenated optimization for parameter estimation, was proposed for the regression of data points acquired with multiple flip angles. The initial estimation of T1 was obtained from the linear regression, followed by the constrained nonlinear regression based on the initial estimates. This heterogeneous initialization strategy improves the fitting accuracy and reduces the computational time. A computationally efficient implementation of concatenated optimization for parameter estimation was achieved based on the graphic processing unit, named as concatenated optimization for parameter estimation graphic processing unit. In experimental comparison with Fram’s method and the Fitter Tool in Jim, the proposed methods are capable of achieving significantly higher efficiency and more accurate estimations. Magn Reson Med 63:1431– C 2010 Wiley-Liss, Inc. 1436, 2010. V Key words: fast low angle shot; FLASH; magnetic resonance imaging; MRI; Ernst formula; longitudinal relaxation time
T1 is an important intrinsic biophysical property of biologic tissues. The T1 values are useful not only in tissue characterization for the diagnosis of pathology such as multiple sclerosis (1) but also in dynamic contrast agent MRI to monitor tumor growth and to assess treatment (2). The original fast low angle shot (FLASH) sequence was proposed by Haase et al. (3). It is now widely used in clinical application such as the three-dimensional acquisition of the brain at very high spatial resolution (4), the imaging within a single breath hold (5,6), and dynamic imaging of the beating heart (7). The signal from a FLASH sequence with a flip angle a follows the Ernst formula (8,9), which represents the signal intensity
1 Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong 2 Department of Diagnostic Radiology and Organ Imaging, The Chinese University of Hong Kong, Shatin, NT, Hong Kong Grant sponsor: Innovation and Technology Fund of the Hong Kong Special Administrative Region, China; Grant number: ITS/042/09. *Correspondence to: Lin Shi, PhD, Department of Diagnostic Radiology and Organ Imaging, The Chinese University of Hong Kong, Shatin, NT, Hong Kong. E-mail:
[email protected] Received 12 March 2009; revised 16 September 2009; accepted 28 October 2009. DOI 10.1002/mrm.22294 Published online in Wiley InterScience (www.interscience.wiley.com). C 2010 Wiley-Liss, Inc. V
as a function of a and other two parameters, namely, the proton density (M) and the T1, given that the echo time value is greatly less than T2*. Research in the estimation of T1 map in FLASH is still active (10,11). The most popular method for T1 estimation uses data points from two flip angles for computational efficiency, and researchers are interested in developing even faster T1 estimation methods (11–15). Due to errors in the pulse flip angles, however, the gradient echo sequence is particularly sensitive to systematic errors (16). Therefore, the T1 map estimated from multiple flip angles (>2) is essentially more accurate than the two-point estimation (2). The method for T1 map estimation using multiple flip angles was first proposed by Fram et al. (17), and it has been conventionally applied in clinical diagnosis. In the Fram et al. (17) paper, the Ernst formula was reformulated as a linear regression problem, but the use of linearization alters the essential meaning of the original objective function, which subsequently influences the estimated parameters values. Jim1 (Xinapse Systems Ltd, Fullers Close, Aldwincle, UK) is commercial software for medical image analysis. The Fitter Tool in Jim v5.0 performs nonlinear least squares regression (fitting) on a series of images that represent different values of a variable. The Fitting Tool can produce the map of T1 values, given the Ernst formula and FLASH images with multiple flip angles as input. In this study, we aim to propose a robust and fast algorithm to accurately estimate the parameters of T1 and M from a sequence of FLASH MRI scans with multiple (2) flip angles. Estimation of these two parameters was formulated as a constrained nonlinear regression problem, where the constraints guarantee that the solution is reasonable and robust. To ensure the nonlinear optimization problem converges to a good estimation robustly and efficiently, the solution of the linear regression was utilized as the initial estimate of the nonlinear regression in our formulation. As the linear regression and the nonlinear regression were concatenated, we named this novel method concatenated optimization for parameter estimation (COPE). To further shorten the execution time of COPE, we adopted the powerful parallel computation technique enabled by the state-of-the-art graphic processing unit (GPU) (18), and this GPU-accelerated COPE was named as COPE-GPU.
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MATERIALS AND METHODS
Table 1 Imaging Parameters of the Six Subjects
Human Subjects and Equipment The test data were acquired from six patients with tumor(s) present in the head and neck. The study was formally approved by the university and hospital ethics committee, and informed written consent was obtained from the patients. The data of two subjects were acquired by a 1.5-T Philips Gyroscan ACS-NT clinical whole-body MRI system (Philips Medical Systems, Best, The Netherlands), and the other four subjects were scanned using a 3.0-T Philips MRI System (Achieva, X Series, Quasar Dual). In our experiment, we used a personal computer with Pentium IV processor, 2.0-GHz CPU, 3.0-GB RAM, and NVIDIA GeForce 8400 GS display adaptor. The program of the Fram’s method was downloaded from the website of course Psych221 in Stanford University2 and translated from Matlab to Cþþ language. The execution configuration of GPU for each block is 128 threads in COPE-GPU. The programming language used in Jim is Java, in Fram’s method and COPE is Cþþ, and in COPE-GPU is Compute Unified Device Architecture (NVIDIA Corporation, Santa Clara, CA, USA) (18). Data Acquisition Following conventional MRI, the patient underwent MRI using a Syn head neck coil with a fast field echo sequence. The details of the imaging parameters are reported in Table 1. Data Analysis Ernst formula (8,9) representing the signal intensity S(a) at a certain flip angle a is formulated as SðaÞ ¼ M sinðaÞ
1 eðTR=T 1 Þ ; ð1 cosðaÞ eðTR=T1 Þ
½1
where M denotes the proton density and T1 is the longitudinal relaxation time, assuming echo time T2*.
Subject ID
Imaging parameters 1.5 T; TR ¼ 2.7 ms; TE ¼ 1 ms; resolution ¼ 128 128 25; flip angle ¼ 2 , 10 , 20 , 30 1.5 T; TR ¼ 2.7 ms; TE ¼ 1 ms; resolution ¼ 128 128 25; flip angle ¼ 2 , 10 , 20 , 30 3.0 T; TR ¼ 4 ms; TE ¼ 1.07 ms; resolution ¼ 128 128 25; flip angles ¼ 2 , 7 , 15 3.0 T; TR ¼ 4 ms; TE ¼ 1.67 ms; resolution ¼ 240 240 20; flip angles ¼ 5 , 10 , 15 3.0 T; TR ¼ 5.42 ms; TE ¼ 1.65 ms; resolution ¼ 240 240 20; flip angles ¼ 5 , 10 , 15 3.0 T; TR ¼ 4 ms; TE ¼ 1 ms; resolution ¼ 128 128 25; flip angles ¼ 2 , 7 , 12 , 15
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TE, echo time; TR, pulse repetition time.
tive is to minimize Pn ^in2 this linearized formulation ^2 i¼1 di , where the squared error di between the estiSðai Þ mated value E tanða þ M ð1 EÞ and the measured iÞ Sðai Þ value sinðai Þ for the ith data point is ^2 ¼ d i
2 Sðai Þ Sðai Þ E M ð1 EÞ : sinðai Þ tanðai Þ
½4
In this linearization, it is easy and fast to calculate the T1 map, but the accuracy is reduced as the essential meaning of the original objective function (i.e., Eq. 5) is altered, which subsequently influences the estimated parameters values. In addition, the values of T1 and M may be negative because there is no any constraint for the linear regression. The estimated values from this linearized regression, however, can be used as good initial guesses for the subsequent nonlinear regression. COPE: Step 2: Nonlinear Regression
COPE: Step 1: Linear Regression Following the method described in Fram et al. (17), Eq. 1 was reformulated as a linear regression problem, as follows, SðaÞ SðaÞ ¼E þ M ð1 EÞ; sinðaÞ tanðaÞ
½2
E ¼ eðTR=T1 Þ :
½3
The second step of the concatenated optimization is to resolve the original nonlinear regression formulated in Eq. 1. The squared error d2i between the estimated value and the measured value for the ith data point is di2 ¼
Sðai Þ
where
Using linear regression (19), the slope a and intercept b can be estimated, from which T1 and M can be obtained, i.e., TR T1 ¼ ; ln a
b and M ¼ : 1a
Suppose (ai, S(ai))ni¼1 represents n pairs of flip angle ai and the corresponding signal intensity S(ai). The objec2
http:/scien.stanford.edu/class/psych221/projects/06/cukur/T1_reconstruction.m
M sinðai Þ ð1 E Þ 2 : 1 cosðai Þ E
½5
The optimal estimates of M and E can be obtained by solving a constrained nonlinear least squares problem, as follows, min M;E
s:t:
Pn i¼1
di2
0
