Relaxation times define the rate of spin magnetic equilibrium recovery in nuclear magnetic resonance (NMR) [1,2]. For each tissue, several relaxation times can be defined. Besides, the main interest is in the evaluation of two of them: the spin-lattice and the spin-spin relaxation times, commonly referred to as T₁ and T₂, respectively. Such time constants, together with the hydrogen nuclei abundance, ρ, define the behavior of the signal generated by each resolution element.

It is largely known that the knowledge of relaxation times can provide interesting information about imaged tissues. Concerning the medical diagnostic field, many pathologies have been found to involve a significant variation of the relaxation time constants more than a variation of ρ, such as Alzheimer's disease [3], Parkinson's disease [4] and cancer [5,6]. The evaluation of the tissue relaxation times can be considered an excellent tool for improving clinical diagnosis.

Classic approaches for retrieving relaxation parameter maps of imaged tissue slices propose the estimation of T₁ and T₂ separately. In particular, the “gold standard” for spin-lattice relaxation time T₁ estimation exploits inversion recovery (IR) sequences [7,8]. However, this approach is too slow for in vivo clinical applications. Different evolutions have been proposed in the literature. In particular, the exploitation of spoiled gradient-recalled echo (SPGR) sequences has shown interesting results [9,10]. With respect to spin-spin relaxation time T₂ estimation, a widely used imaging sequence is the spin echo (SE) [11,12].

The magnitude of the acquired signal is typically used for relaxation parameter estimation [12–15]. Within this framework, the exponential curve fitting via the least squares (LS) algorithm is the commonly adopted estimator [11,13]. Although being very easy to be implemented and not computationally heavy, it has the disadvantage of producing biased estimations [11,16]. The alternative consists in using a maximum likelihood estimator (MLE) [12]. The MLE is asymptotically unbiased and optimal, but the function to be maximized, which is related to the statistical distribution of the MRI amplitude data, is computationally heavy, as it contains the Bessel function [17].

Recently, new approaches based on the complex decomposition of acquired data have been proposed [10,18]. The exploitation of the complex model leads to a main advantage concerning the estimation: due to the circular Gaussian distribution of the complex noise, the LS-based estimator coincides with the MLE and is asymptotically unbiased and optimal.

While much effort has been directed to improving the estimation procedures, only a little effort has been directed to the choice of the optimal imaging parameter selection (i.e., the optimal choice of the MRI scanner imaging parameters). In particular, in [19], the ideal repetition times have been investigated in the case of saturation recovery spin-lattice measurements at 4.7 T, while in [20], the optimization of T₂ measurements in the case of bi-exponential systems is considered. Following the approach proposed by [15], within this paper we investigate the possibility of finding the optimal imaging parameter configuration for relaxation time estimation. As an alternative to [15], we investigate the optimal configuration not only for the T₂ time estimation, but for the joint T₂ and T₁ estimation.

Since the SE sequence-based model allows the simultaneous estimation of both spin-spin and spin-lattice relaxation times, we focus our attention on this imaging sequence. In any case, the theoretical study reported in the following could be easily adapted to different imaging sequences. Considering an SE sequence [2], the two imaging parameters involved in the acquisition procedure are the repetition time, T_R, and the echo time, T_E. We briefly recall that these two parameters allow the scanner to differently interact with tissues characterized by different T₁ and T₂ values. By exploiting different T_R and T_E combinations, it is possible to emphasize the effect of one tissue intrinsic parameter with respect to others, obtaining the well-known ρ-weighted, T₁-weighted or T₂-weighted images.

Given the previously mentioned motivations, we present a deeper analysis of the complex SE model considered in [18] extended to three parameters (i.e., ρ, T₁ and T₂). The analysis is conducted exploiting the Cramer–Rao lower bounds (CRLBs) [16]. Since CRLBs provide the best achievable performances in the unbiased estimation of one or more parameters, by minimizing them with respect to the MR scanner imaging parameters, it is possible to find the optimal acquisition configuration for the relaxation time estimation. Practically speaking, we look for the acquisition parameters that allow achieving lower relaxation time estimation errors. The result of the study is the introduction of a general empirical rule for determining the optimal (with respect to CRLBs) MRI scanner parameter configuration. In particular, the identification of these parameters in the case of several tissues has been conducted. The effectiveness of the theoretical results and of the empirical rule is validated and verified on different datasets.

The manuscript is organized as follows: in Section 2, the acquisition model for an MRI spin echo sequence is presented, and in Section 3, the achievable performances of the estimation are analyzed via the CRLBs. In Section 4, the CRLB-based empirical rule for the optimal acquisition parameter configuration is presented. Finally, validation on different datasets is presented in Section 5, and conclusions are drawn.

Let us consider an MRI acquisition system using a spin echo imaging sequence. The amplitude of the recorded complex signal after the image formation process, i.e., after the computation of the 2D Fourier transform, is related to the tissue parameters, ρ, T₁ and T₂, via [2,21]: (1) f ( θ ) = ρ exp ( − T E T 2 ) ( 1 − exp ( − T R T 1 ) )where T_E and T_R are the echo and repetition time, respectively, which are two imaging parameters that can be set in the MRI scanner, and θ = [ρ T₁ T₂]^T is the vector containing the tissue parameters in which we are interested. Note that Equation (1) is valid in the case of a homogeneous imaged object. In the case of clinical data, the presence of different hydrogen environments within each voxel has to be taken into account. The acquisition model reported in Equation (1), which is a solution to Bloch equations, assuming that T_E is short with respect to T_R, is related to the noise-free case and does not take into account the dependency on the static magnetic field, B. Considering noise, in the complex domain, the model becomes: (2) y = y R + i y I = f ( θ ) exp ( i ϕ ) + ( n R + i n I )where n_R and n_I are the real and imaginary parts of the noise samples, which are distributed as independent circularly Gaussian variables [22], and ϕ represents the angle of the complex data [23,24]. Thus, the statistical distributions of the real and imaginary parts of the acquired signal are: f Y R ( y R ) = 1 2 π σ 2 exp ( − ( y R − f ( θ ) cos ( ϕ ) ) 2 2 σ 2 ) (3) f Y I ( y I ) = 1 2 π σ 2 exp ( − ( y I − f ( θ ) sin ( ϕ ) ) 2 2 σ 2 )where σ² is the variance of real and imaginary noise components. Due to the independence of the real and imaginary parts of noise, the joint statistical distribution of y_R and y_I is the product of the two probability density functions of Equation (3).

Once N acquisitions with different T_R and T_E combinations have been recorded and collected in the data vector y = [y_R, y_I], with y_R = [y_R(1), ⋯ y_R(N)] and y_I = [y_I(1), ⋯ y_I(N)], we can derive the likelihood function from the factorization of the Probability Density Functions (PDFs): (4) p ( y ; θ ) = ∏ k = 1 N ( 1 2 π σ 2 ) 2 exp { − [ y R ( k ) − f ( θ ) cos ( ϕ ) ] 2 2 σ 2 − [ y I ( k ) − f ( θ ) sin ( ϕ ) ] 2 2 σ 2 }

Starting from the likelihood function of Equation (4), the CRLBs for θ are derived and analyzed in the following sections.

In order to evaluate the performances of the optimal estimator for the model presented in Section 2, the Cramer–Rao lower bounds have to be computed. According to Statistical Estimation Theory [16], given an observation model, the accuracy of any estimator can be evaluated according to its mean and its variance. In particular, in order to be optimal, an estimator needs to have its mean equal to the value to be estimated (i.e., unbiased estimator) and to have the smallest possible variance. CRLBs represent the lower bound of the variance of any unbiased estimator, resulting an interesting and powerful tool for evaluating the achievable performances of a considered model. By computing the CRLBs for different configuration of the parameters involved in the acquisition model, it is possible to find the best parameter configuration, the one that provides the minimum values of CRLBs (i.e., the minimum achievable variances). Considering the vector parameter θ, the minimum variance that any unbiased estimator of parameter θ_i can reach is provided by the i-th diagonal element of the inverse of matrix I [16]: (5) var ( θ ^ i ) ≥ [ I − 1 ( θ ) ] i iwith I being the Fisher matrix, which is equal to: (6) I ( θ ) = [ − E [ ∂ 2 ln p ( y ; θ ) ∂ ρ 2 ] − E [ ∂ 2 ln p ( y ; θ ) ∂ ρ ∂ T 1 ] − E [ ∂ 2 In p ( y ; θ ) ∂ ρ ∂ T 2 ] − E [ ∂ 2 ln p ( y ; θ ) ∂ ρ ∂ T 1 ] − E [ ∂ 2 ln p ( y ; θ ) ∂ T 1 2 ] − E [ ∂ 2 ln p ( y ; θ ) ∂ T 1 ∂ T 2 ] − E [ ∂ 2 ln p ( y ; θ ) ∂ ρ ∂ T 2 ] − E [ ∂ 2 ln p ( y ; θ ) ∂ T 1 ∂ T 2 ] − E [ ∂ 2 ln p ( y ; θ ) ∂ T 2 2 ] ]where E [·] is the expected value operator.

A closed form for the second order derivatives of Equation (6) has been derived and reported in the Appendix. The closed form greatly improves the computational accuracy of the CRLB evaluation and decreases the computational burden of the simulations reported in the following.

In order to experimentally compute the matrix of Equation (6), Monte Carlo simulations with 10⁵ iterations have been considered for statistical average computation.

For the following evaluations, we considered a tissue, named A, with parameters θ = [ρ T₁ T₂]^T = [2.5 1600 90]^T. Note that within the paper, all relaxation times are expressed in milliseconds, while the proton density is in percentage. The following simulations are reported and analyzed in order to investigate CRLB dependency and behavior with respect to the signal-to-noise ratio (SNR), the number of acquisitions and the scanner acquisition parameters.

After the evaluation of ρ, T₁ and T₂ CRLB behaviors, an analysis dedicated to the computation of optimal T_R and T_E combinations is presented. In the following, it will be shown that a proper imaging configuration can greatly improve the performances with respect to such a choice. In particular, the aim of this section is to identify the ideal imaging parameters with respect to imaged tissues.

Let us show how the optimal imaging parameters can be determined. Initially, we have focused on the minimization of T₂ CRLB, which consist in finding T_E values that minimize the element (3, 3) of the inverse of the Fisher matrix, I(θ), of Equation (6) for different spin-spin relaxation times T₂. The optimization has been performed by searching the three optimal T_E values in the [82, 350] ms range for a fixed value of T_R. The evaluation has been done varying the tissue T₂ relaxation times in the [20, 200] ms range, obtaining the results shown in Figure 5. Some considerations can be drawn:

there is no T_E value combination that is simultaneously ideal for tissues with different spin-spin relaxation times. As a consequence, we can only find the T_E combination that is ideal for a specific tissue;

From these simulations, we can derive an empirical rule for the optimal T_E selection: the lower one should be fixed to the minimum value accepted by the MR scanner, while the other values should to be set in the range of 100%–120% of the value of (T_E(1) + T₂), considering the T₂ of the tissue in which we are interested.

A similar evaluation has been conducted for the minimization of T₁ CRLB varying MRI scanner repetition times T_R, with a fixed value of T_E; the results are shown in Figure 6. The higher T_R value is fixed to the right edge of the considered variability range, which we set equal to 4, 000 ms. The intermediate and low T_Rs have similar values, which, starting from 500 ms in the case of tissue with T₁ = 700 ms, grow almost linearly up to 1, 400 ms for tissues with higher T₁ (about 3, 000 ms). It is hard to determine an empirical rule in this case; anyway, we can say that a choice of around 1, 000 ms for T_R(1) and T_R(2) will fit a wide class of tissues, i.e., those with 1, 200 < T₁ < 2, 000 ms.

Concluding this section, one more evaluation has been conducted. Instead of optimizing T_R and T_E values separately, a joint minimization has been done. Nine acquisitions have been considered, related to three repetition and three echo times. Among the three values, the lower and the higher have been fixed to the search range bounds, so only the intermediate T_E and T_R values were variable. Results are shown in Figure 7, respectively. It is evident from the figure that T_E values can be considered independent from T₁, as far as T_R from T₂, proving the correctness of the separate optimization of the echo and repetition times. In particular, from Figure 7a, we can state that tissues with equal T₂, but very different T₁ values share the same three optimal echo times for T₂ estimation, and vice versa. That said, the behaviors of Figures 5 and 6, i.e., the minimization, one parameter at a time, are confirmed.

In order to easily apply the obtained results, the ideal acquisition parameters for different tissues have been computed exploiting CRLB minimization in the case of a 1.5 T and a 3 T MRI scanner. The results are shown in Tables 1 and 2 for T₁ and T₂, respectively. According to the results reported in Figure 7, the minimizations have been computed independently for spin-lattice and spin-spin relaxation time estimation. Tissue relaxation times have been simulated according to reference values present in the literature [25], which are reported in Table 3.

In Table 4, optimal echo times in the case of gray matter T₂ estimation for different minimum T_E are reported. It can be noticed that the lower optimal echo time is always the minimum and that the empirical rule is confirmed.

Note that the usefulness of a proper T_R and T_E selection, besides the lower estimation variance, consists also in reducing the acquisition time. In order to make such an advantage evident, Table 5 reports the achievable performance in the case of 16 images (4 T_R and 4 T_E values) when moving from equally spaced to optimized acquisition parameters. In particular, the last column of Table 5 shows that 12 acquisitions, with properly chosen parameters, can lead to better results with respect to 16 equally spaced images, while definitely reducing the global acquisition time.

Within this section, some numerical results are shown in order to validate the advantage of the optimal selection of the imaging parameters according to the previously reported theoretical studies. A tissue with parameters [ρ T₁ T₂] = [5.5 775 44.5] has been considered. Three noisy datasets (SNR = 30 dB) have been simulated, each one composed of four acquisitions. The parameters of Dataset 1 have been chosen according to the results of Figures 5 and 6 in order to optimize the estimation for the considered tissue. Datasets 2 and 3 have been generated with non-ideal parameters. The dataset characteristics are summarized in Table 6.

To asses and validate the CRLB studies, the estimation of the relaxation times has been implemented via Monte Carlo simulation. In particular, a maximum likelihood estimator (MLE) has been set up in the complex domain. Considering that the noise is circularly Gaussian distributed, MLE corresponds to a non-linear least squares (NLLS) estimator [18]. It is important to note that the previously reported theoretical studies about the optimal selection of the imaging parameters are valid for any unbiased estimator, since CRLBs are related only to the acquisition model. Among different estimators, NLLS has been chosen thanks to its low computational times and complexity. We recall that the choice of the optimal estimator is not the aim of this paper.

The NLLS estimator for the ρ, T₁ and T₂ parameters has been implemented on the three datasets of Table 6. A quantitative analysis of the results, in terms of estimation means and variances, has been reported in Table 7. By analyzing it, it is possible to infer that the estimator means are very close, while variances significantly vary from one dataset to the other. In particular, the smallest variances are obtained in the case of Dataset 1. This fully agrees with the theoretical studies reported in Section 4; as a matter of fact, Dataset 1 has been generated by using the previously developed optimal T_E and T_R parameter selection for the considered relaxation times. It is evident that choosing a non-ideal imaging parameters configuration can lead to very inaccurate results. For example, the T₂ estimator variance of Dataset 3 is approximately six times larger than the one of Dataset 1. In order to visualize such results, the normalized standard deviations of ρ, T₁ and T₂ in the case of Datasets 1, 2 and 3 are reported in Figure 8.

The higher achievable accuracy in the case of optimal imaging parameters selection can also be inferred from the empirical probability density functions of the estimators, reported in Figure 9. In each image, the blue, the green and the red curves refer to Datasets 1, 2 and 3 of Table 6. As expected, most of the presented estimators follow a Gaussian distribution, with a different width. Blue curves obtained using Dataset 1, characterized by the optimal T_R/T_E values for the simulated tissue, are always the narrowest (smallest variances). Moving to curves obtained from Datasets 2 and 3, the estimation error becomes larger. Moreover, in the case of the ρ estimator, the results start showing a bias in the case of Dataset 3, i.e., the one with the worst acquisition parameters, and the empirical PDF does not look like a Gaussian function any more.

Finally, one further simulation is presented. Signals from two different tissues have been simulated, with parameters [ρ T₁ T₂] = [5 700 68] (spinal cord) and [ρ T₁ T₂] = [5 1190 115] (gray matter). Two datasets composed of four acquisitions have been generated, with parameters reported in Table 8. Taking into account the developed procedure, Dataset 1 parameters represent the ideal configuration for the first tissue, while Dataset 2 is the ideal for the second one.

The empirical probability density functions for T₁ and T₂ estimators have been computed for both datasets and are shown in Figures 10, respectively. Once again, The results validate the theoretical study of Section 4. Estimation based on Dataset 1 (red line) shows lower variance in the case of spinal cord, i.e., the tissue with the lowest relaxation times (the left peaks of Figures 10). Considering gray matter, Dataset 2 (blue line) -based estimation gives better results, although the improvement of the T₁ estimator is not pronounced. Once again, the result highlights the need of properly tuning the acquisition parameters.

Within this paper, an analysis on the spin echo signal model in MR imaging has been addressed. In particular, Cramer–Rao lower bounds for relaxation time estimation in the case of a complex Gaussian acquisition model have been evaluated. Several CRLB-based evaluations have been presented in order to investigate the possibility of finding the optimal, in terms of reconstruction accuracy, imaging parameter configuration for the estimation of T₁ and T₂ maps. According to these theoretical studies, an empirical rule together with the identification of the optimal imaging parameter combination (echo and repetition times) in case of different tissues (different T₁ and T₂) has been proposed. Moreover, the optimal acquisition parameters for several tissues have been computed for both 1.5 T and 3 T acquisitions. The theoretical results have been numerically validated on different datasets. It should be underlined that such optimal parameters are independent from the implemented estimators, as CRLBs only depend on the signal model. Once the data have been acquired, different estimators proposed in the literature can be applied. It is important to underline that the theoretical studies reported within the paper can be easily adapted to different imaging sequences.