Let us consider a pixel of a complex-valued MR image; in a noise free case, the signal expression is given by:y0=mei?(1)where m represents the amplitude of the signal and ? the phase. The amplitude m depends on the used imaging sequence Gemcitabine synthesis and on the unknown parameters �� (the spin density of hydrogen atoms ��, the spin-lattice relaxation time T1 and the spin-spin relaxation time T2) via:m=f(��)=f(��,T1,T2)(2)while the phase ? depends on the local intensity of the magnetic field (field map) [16][17].Equation (1) can be alternatively written in terms of real and imaginary parts:y0=y0R+i?y0I=mcos(?)+i?msin(?)=f(��)cos(?)+i?f(��)sin(?)(3)Let us now focus on the noisy case.
As stated before, real (yR) and imaginary (yI) parts are corrupted by additive, zero mean and uncorrelated Inhibitors,Modulators,Libraries Gaussian noise wR and wI:yR=f(��)cos(?)+wRyI=f(��)sin(?)+wI(4)providing the following distributions:PYR(yR)=12��2e?(yR?f(��)cos(?))22��2PYI(yI)=12��2e?(yI?f(��)sin(?))22��2(5)where ��2 represents noise variance. By multiplying the two equations (5), we obtain the joint statistical distribution The proposed method is based on this statistical Inhibitors,Modulators,Libraries distribution of the measured data, called Gaussian Complex Model.Other techniques work on the amplitude instead Inhibitors,Modulators,Libraries of the complex domain, assuming different distribution for the data [14]. In particular, starting from Equation (5), it is easy to show that the amplitude is corrupted by Rice distributed noise [5�C7], leading to the following distribution:PM(m^)=m^��2e(?m^2+f2(��)2��2)I0(f(��)m^��2)(6)where I0(?) is the modified Bessel function of the first kind with order zero and represents the measured noisy amplitude.
The term f(��) is determined by the imaging sequence adopted during the MR scan. In this paper we consider a conventional spin echo imaging sequence which leads to [2]:f(��)=��?e?TET2?(1?e?TRT1)(7)where the instrumental variables TE and TR are the Echo Time and the Repetition Time of the sequence, respectively.Since we are interested in T2 estimation, we enclose T1 and �� in a Inhibitors,Modulators,Libraries new parameter k, called pseudodensity, as in [13]. So f(��) becomes:f(��)=k?e?TET2(8)which is commonly referred as monoexponential transversal magnetization decay model and is widely adopted [13,14]. In the next Section the achievable accuracy for T2 estimation using the proposed model is addressed exploiting CRLB.3.
?Cramer Rao Lower Bounds EvaluationThe Cramer Rao Lower Bounds AV-951 provide the minimum selleck chemical variance for a given acquisition model that any unbiased (non polarized) estimator can reach [18]. They provide a benchmark against which we can compare the performances of any unbiased estimator. Moreover, they alert us of the physical impossibility of finding an unbiased estimator with a variance smaller than the bounds. CRLB for amplitude estimation in MR Imaging have been presented in different papers [8,9]. In this section, a study on the CRLB for the specific T2 estimation problem using the Gaussian Complex Model is conducted.