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Magnetic Resonance (MR) imaging techniques are used to measure biophysical properties of tissues. As clinical diagnoses are mainly based on the evaluation of contrast in MR images, relaxation times assume a fundamental role providing a major source of contrast. Moreover, they can give useful information in cancer diagnostic. In this paper we present a statistical technique to estimate relaxation times exploiting complex-valued MR images. Working in the complex domain instead of the amplitude one allows us to consider the data bivariate Gaussian distributed, and thus to implement a simple Least Square (LS) estimator on the available complex data. The proposed estimator results to be simple, accurate and unbiased.

Magnetic Resonance Imaging (MRI) is a technique used in medical environment to produce high quality images of tissues inside human body. MRI is based on the principles of nuclear magnetic resonance (NMR), a spectroscopic technique used to obtain microscopic chemical and physical information about molecules. In MRI, a radiofrequency signal stimulates the hydrogen atoms of an object posed in a uniform magnetic field to emit complex-valued signals. These signals are coherently collected by MR coil, converted from analog to digital and recorded in the so called

MR images are characterized by the combination of three intrinsic parameters: the spin density of hydrogen atoms _{1} and the spin-spin relaxation time T_{2} [_{2} can give useful information for cancer discrimination [

Conventional relaxation parameter estimation techniques work on the amplitude of MR images. A commonly used approach consists in using Least Square (LS) estimator [_{2} estimation problem. In the first one [

In this paper we propose an approach for spin-spin relaxation time estimation that works directly on the complex-valued MR images instead of amplitude. Working in complex domain allows us to implement LS estimation since the noise is Gaussian both on real and imaginary parts. Differently from [

In Section 2 the statistical model is briefly addressed. In Section 3 the accuracy of the proposed model is evaluated exploiting Cramer Rao Lower Bounds (CRLB) and a comparison with other models present in literature is discussed. The performances of the proposed estimator are shown in Section 4. In Section 5, a fast version of the Non Linear LS estimator is proposed. Finally, we draw some conclusions about the presented technique.

In MRI the data are recorded in the

Let us consider a pixel of a complex-valued MR image; in a noise free case, the signal expression is given by:
_{1} and the spin-spin relaxation time T_{2})

Let us now focus on the noisy case. As stated before, real (_{R}_{I}_{R}_{I}^{2} represents noise variance. By multiplying the two equations

Other techniques work on the amplitude instead of the complex domain, assuming different distribution for the data [_{0}(·) is the modified Bessel function of the first kind with order zero and

The term _{E}_{R}

Since we are interested in T_{2} estimation, we enclose T_{1} and _{2} estimation using the proposed model is addressed exploiting CRLB.

The Cramer Rao Lower Bounds provide the minimum variance for a given acquisition model that any unbiased (non polarized) estimator can reach [_{2} estimation problem using the Gaussian Complex Model is conducted. To assess the performances of the proposed model, the CRLB are numerically evaluated using Monte Carlo method.

In the following simulation, a pixel with _{2} = 100 msec is considered, corresponding to white matter tissue at 1.5 Tesla. The number of Monte Carlo iterations is fixed to 10^{5}.

First of all the dependency of CRLB on SNR is analyzed. SNR is defined as the ratio between the mean value of _{E}

Investigating the behavior of CLRB in

The dependency of CRLB on the number of available acquisition (_{E}_{E}

Investigating the behavior of CLRB in

The CRLB evaluation can be useful not only to compare the achievable accuracy of different models, but also to investigate the best parameter configuration for a considered model.

In order to evaluate the impact of T_{E}_{E}_{E}_{E}_max_{E}_max_{E}_max_{E}_{E}_max

Finally, the influence of phase _{E}

In this section, we present the estimator for the proposed model and we evaluate its performances in terms of mean and variance, compared to the CRLB, for different case studies.

Since we are working in the complex domain, as previously stated, signal statistical distributions for both real and imaginary parts are Gaussian. This assumption allows us to use a simple Least Square estimator. In particular we implement a Non Linear LS (NLLS) as the signal is characterized by an exponential decay

The presented results are obtained applying the method to realistically simulated data sets, using the same parameters of the previous section.

In the first case study the performances of the proposed estimator for different number of available complex-valued data sets _{2} estimator for different number of T_{E}_{2} estimator, compared to the CRLB. It can be noted that the estimator rapidly tends to be efficient, since its variance approaches the CRLB as the number of data grows.

For the second case study, we consider the performances of the T_{2} estimator for different values of SNR. We fix the number of acquisitions to 8 (

Finally the proposed algorithm has been tested on a 256 × 256 realistically simulated slice of the human head. The spin density and the relaxation times of the simulated tissues are reported in _{R}

_{E}

The results of the ML estimator applied to the Rician amplitude model and of the NLLS estimator with Gaussian complex model are reported in

Concerning the computational cost of the algorithm, we have to underline that an optimization algorithm needs to iteratively compute a function to find the solution. Clearly, Gaussian Complex Model (

The better accuracy of the Gaussian Complex Model respect to the Rician one can be explained in terms of functions to be optimized. In the first case, the square error function between the observed data and

In Section 3 we presented the comparison between the Amplitude/Rice and Complex/Gaussian models, showing the best performances of the second one in both estimation accuracy and computational time. In this section we introduce a method to improve the speed of the second approach, making it quasi real time. This approach is based on Non Linear LS and is defined in [_{1}_{R}_{1}_{I}_{2}_{R}_{2}_{I}_{NR,} y_{NI}^{T} is the collection of the real and imaginary parts of measured data for the _{E}_{E}_{1}, T_{E}_{2}_{EN}^{T}, and:
_{2}) are reported twice in order to take into account both real and imaginary parts of

The estimator of _{E}

As we can see, in terms of performances, the fast estimator can be considered unbiased as the classic one while the accuracy is slightly worst compared to the classic one. This is due to the non linear relation between the observation matrix _{2} in equation

A final note about the possibility of estimating T_{1} has to be marked. In the proposed approach we are interested in the estimation of T_{2}. Thus all our simulations and analysis are conducted considering only the spin-spin relaxation parameter. Anyway, it is easy to show that the proposed approach can be extended also for the estimation of T_{1}, expliciting

In this paper a new statistical technique for the estimation of relaxation time parameters in Magnetic Resonance Imaging is presented. Differently from other models present in literature, we propose to work directly with real and imaginary parts of complex-valued Magnetic Resonance images. This implies to consider the acquired data corrupted by additive, zero mean, uncorrelated complex-valued Gaussian noise samples, allowing us to use a simple Non Linear LS estimator to retrieve the unknown parameter T_{2} without introducing any bias in the estimation. The evaluation of CRLB let us state that the proposed model is able to reach a better accuracy compared to other models. Moreover, the conducted simulations show the good performances of the proposed Non Linear LS estimator: the estimator is unbiased and rapidly tends to be efficient. Finally a fast Non Linear LS estimator has been proposed, providing an algorithm for quasi real time applications. The next step of this work will be the application of the approach to real data sets.

_{0}field variations

CRLB behavior in case of different SNRs for the proposed Gaussian Complex Model (blue) and for the Rice Amplitude Model (red).

CRLB behavior in case of different number of acquisitions for the proposed Gaussian Complex Model (blue) and for the Rice Amplitude Model (red).

Impact of different spacing between TE values on the CRLB.

Impact of

Performances of the T_{2} estimator for different number of available complex-valued data. (a) Mean behavior (the mean is plotted in blue and the true value is plotted in red) and (b) variance behavior (the variance is plotted in blue and the CRLB are plotted in red). The results are obtained using SNR = 4, _{E}

Performances of the T2 estimator for different SNRs. (a) Mean behavior (the mean is plotted in blue and the true value is plotted in red) and (b) variance behavior (the variance is plotted in blue and the CRLB are plotted in red). The results are obtained using

Reference spin density map (a), reference T_{1} map (b), reference pseudodensity map (c), reference T_{2} map (d).

Estimated pseudodensity map using ML estimator with Rician amplitude model (a), estimated T_{2} map using ML estimator with Rician amplitude model (b), estimated pseudodensity map using LS estimator with Gaussian complex model (c), estimated T_{2} map using LS estimator with Gaussian complex model (d), T_{2} reconstruction error map using ML estimator with Rician amplitude model (Normalized mean square error equal to 0.1546) (e), T_{2} reconstruction error map using LS estimator with Gaussian complex model (Normalized mean square error equal to 0.1029) (f).

Estimated T_{2} map using LS estimator with Gaussian complex model in case of _{2} reconstruction error map (Normalized mean square error equal to 0.0210) (b).

Behaviors of normalized functions to be minimized in case of Gaussian Complex Model (a–c) and Rician model (e–f) for SNR = 1 (a,d), SNR = 3 (b,e) and SNR = 5 (c,f).

Performances of the Fast T_{2} estimator for different number of available complex-valued data. (a) Mean behavior (the mean of Fast Non Linear LS estimator is plotted in blue, the mean of the classic Non Linear LS estimator is plotted in green and the true value is plotted in red) and (b) variance behavior (the variance of Fast Non Linear LS estimator is plotted in blue, the variance of the classic Non Linear LS estimator is plotted in green and the CRLB value is plotted in red). The results are obtained using SNR = 4, _{E}

CRLB values in case of different SNRs for the proposed Gaussian Complex Model and for the Rice Amplitude Model.

SNR | Gaussian Complex Model | Rice Amplitude Model | Relative Difference |
---|---|---|---|

1 | 1570.5 | 2836.1 | +80.6% |

2 | 399.3456 | 520.6266 | +30.3% |

3 | 172.1815 | 199.2489 | +15.7% |

4 | 97.0954 | 107.6980 | +10.9% |

5 | 63.0583 | 67.1536 | +6.5% |

Spin density and relaxation times for the Shepp-Logan phantom. The _{R}

Tissue | T_{1} (msec) |
T_{2} (msec) | ||
---|---|---|---|---|

Scalp | 80 | 324 | 80 | 70 |

Bone | 12 | 533 | 11.99 | 50 |

CSF | 98 | 2,000 | 89.95 | 500 |

Gray Matter | 74.5 | 857 | 74.28 | 100 |

White Matter | 61.7 | 583 | 61.68 | 80 |

Tumor | 95 | 926 | 94.57 | 120 |