SensorsSensors1424-8220Molecular Diversity Preservation International (MDPI)10.3390/s140202182sensors-14-02182ArticleOptimal Configuration for Relaxation Times Estimation in Complex Spin Echo ImagingBaseliceFabio^{1}^{*}FerraioliGiampaolo^{2}GrassiaAlessandro^{1}PascazioVito^{1}
Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope, Napoli 80143, Italy; E-Mails: alessandro.grassia@uniparthenope.it (A.G.); vito.pascazio@uniparthenope.it (V.P.)
Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli Parthenope, Napoli 80143, Italy; E-Mail: giampaolo.ferraioli@uniparthenope.it
Author Contributions The author contributions are substantially equal. In particular, the main contribution of Vito Pascazio, Giampaolo Ferraioli and Fabio Baselice was the methodology development. Moreover, Giampaolo Ferraioli and Fabio Baselice were specifically focused on the numerical implementation. Alessandro Grassia worked both on code implementation and on software simulation tasks.
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
Many pathologies can be identified by evaluating differences raised in the physical parameters of involved tissues. In a Magnetic Resonance Imaging (MRI) framework, spin-lattice T_{1} and spin-spin T_{2} relaxation time parameters play a major role in such an identification. In this manuscript, a theoretical study related to the evaluation of the achievable performances in the estimation of relaxation times in MRI is proposed. After a discussion about the considered acquisition model, an analysis on the ideal imaging acquisition parameters in the case of spin echo sequences, i.e., echo and repetition times, is conducted. In particular, the aim of the manuscript consists in providing an empirical rule for optimal imaging parameter identification with respect to the tissues under investigation. Theoretical results are validated on different datasets in order to show the effectiveness of the presented study and of the proposed methodology.
Magnetic Resonance Imagingcomplex decompositionstatistical signal processingCramerRao lower boundsrelaxation time estimationIntroduction
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 T_{1} and T_{2}, 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 T_{1} and T_{2} separately. In particular, the “gold standard” for spin-lattice relaxation time T_{1} 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 T_{2} 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 T_{2} 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 T_{2} time estimation, but for the joint T_{2} and T_{1} 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 T_{1} and T_{2} 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, T_{1}-weighted or T_{2}-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, T_{1} and T_{2}). 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.
The Model
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, T_{1} and T_{2}, via [2,21]:
f(θ)=ρexp(−TET2)(1−exp(−TRT1))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 T_{1}T_{2}]^{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:
y=yR+iyI=f(θ)exp(iϕ)+(nR+inI)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:
fYR(yR)=12πσ2exp(−(yR−f(θ)cos(ϕ))22σ2)fYI(yI)=12πσ2exp(−(yI−f(θ)sin(ϕ))22σ2)where σ^{2} 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):
p(y;θ)=∏k=1N(12πσ2)2exp{−[yR(k)−f(θ)cos(ϕ)]22σ2−[yI(k)−f(θ)sin(ϕ)]22σ2}
Starting from the likelihood function of Equation (4), the CRLBs for θ are derived and analyzed in the following sections.
Cramer-Rao Lower Bounds Evaluations
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]:
var(θ^i)≥[I−1(θ)]iiwith I being the Fisher matrix, which is equal to:
I(θ)=[−E[∂2lnp(y;θ)∂ρ2]−E[∂2lnp(y;θ)∂ρ∂T1]−E[∂2Inp(y;θ)∂ρ∂T2]−E[∂2lnp(y;θ)∂ρ∂T1]−E[∂2lnp(y;θ)∂T12]−E[∂2lnp(y;θ)∂T1∂T2]−E[∂2lnp(y;θ)∂ρ∂T2]−E[∂2lnp(y;θ)∂T1∂T2]−E[∂2lnp(y;θ)∂T22]]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 10^{5} iterations have been considered for statistical average computation.
For the following evaluations, we considered a tissue, named A, with parameters θ = [ρ T_{1}T_{2}]^{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.
<italic>CRLB</italic> vs. <italic>SNR</italic>
Let us start by computing CRLBs varying the noise standard deviation (i.e., the SNR). Sixteen images have been considered, which refer to the all combinations of four T_{R} and four T_{E} values equally spaced in the intervals [500, 3500] ms and [80, 350] ms, respectively. Note that the lower T_{E} value has been set according to the minimum echo time for the SE sequence accepted by the Philips Achieva 3.0 T, the MR scanner we worked on, while the maximum value of T_{R} has been set in order to limit the global acquisition time. The CRLBs in the case of different SNRs are shown in Figure 1. As expected, the square root of the CRLBs related to all considered parameters decreases with respect to SNR growth, i.e., high SNRs positively affect the estimator performances. In the considered range of SNRs, no saturation appears. Very similar behaviors are obtained varying T_{R} and T_{E} combinations. Globally, it can be stated that SNR linearly affects CRLBs, so in the following the results of each simulation can be easily extended to any SNR configuration.
<italic>CRLB</italic> vs. <italic>the Number of Acquisitions</italic>
A second case study has been conducted in order to evaluate the advantage of increasing the number of acquisitions. Two vectors of T_{R} and T_{E}, of a length of N_{R} and N_{E}, respectively, have been generated in the [500, 3500] ms (for T_{R}) and [80, 350] ms (for T_{E}) intervals. The square root of CRLBs, i.e., the minimum achievable standard deviations, are reported in Figure 2 for ρ, T_{1} and T_{2}, respectively, for different N_{R} and N_{E} combinations. The noise variance has been fixed in order to obtain an SNR of 16 dB for the image with the lowest signal intensity (i.e., T_{R} = 500 ms and T_{E} = 80 ms). It can be noted that the number of T_{R} values mainly affects the achievable performances with respect to T_{1} estimation, while CRLBs of ρ and T_{2} are dependent on the number of both T_{E} and T_{R} values, with a higher dependency on echo times. The results confirm the strict connections between T_{R} and T_{1} and also between T_{E} and T_{2}, as expected from the exponential terms of the SE signal model (Equation (1)). However, it is interesting to stress how the CRLB of ρ is very tightly related to T_{E} values rather than to T_{R} ones.
<italic>CRLB</italic> vs. <italic>T<sub>R</sub> and T<sub>E</sub> Values</italic>
As a further case study, an evaluation of CRLBs with respect to T_{R} and T_{E} values with a fixed number of acquisition has been performed. Four acquisitions have been considered, corresponding to all the combinations of T_{R} = [T_{R}(1), T_{R}(2)] and T_{E} = [T_{E}(1), T_{E}(2)]. CRLBs have been computed while varying T_{R}(1) and T_{E}(2) and considering T_{R}(2)= 3, 500 ms and T_{E}(1) = 80 ms, again in the case of tissue A parameters. Results are reported in Figures 3. Figure 4a shows that the ρ estimation would prefer low T_{R} (1) and high T_{E}(2) values. The behaviors of CRLBs for T_{1} and T_{2} differ remarkably from Figure 4a, as it can be noticed that the estimation of T_{1} is almost unresponsive with respect to T_{E}(2) values, as far as T_{2} estimation with respect to T_{R}(1). In particular, for the estimation of T_{1}, the ideal T_{R}(1) is as low as possible, while the ideal T_{E}(2) for the estimation of T_{2} is between 150 and 250 ms. For this experiment, a second dataset has also been considered: the same simulation has been conducted in the case of a second tissue, named B, with parameters θ = [ρ T_{1}T_{2}] ^{T} = [2.8 1800 60]^{T}, in order to know if the results of Figure 3 are always valid or if they are highly dependent on the considered tissue. The results are reported in Figure 4. It can be noticed that the lower regions remain in the same position, although being increased in value, but for CRLBs of ρ and T_{2}, the ideal T_{E}(2) range reduces to [130, 180] ms. This is mainly due to the lower T_{2} value of tissue B with respect to tissue A. Thus, it can be concluded that the general trend is confirmed, although the position of the global minimum is strictly related to the considered tissue. These two simulations show that the choice of optimal parameters is strictly dependent on the relaxation times of the imaged tissues. In the next section, we investigate the possibility of finding a rule for ideal imaging parameter identification.
Optimal Parameter Configuration
After the evaluation of ρ, T_{1} and T_{2} 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 T_{2} 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 T_{2}. 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 T_{2} 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;
by analyzing Figure 5, it can be noticed that the lowest T_{E} value of the ideal configuration is always equal to the lower bound of the considered variability range, which, in our case, was fixed to 82 ms. As stated before, this value is the minimum echo time for the SE sequence accepted by the Philips Achieva 3.0 T, the MR scanner we worked on;
the two higher T_{E} values, which are the red and the green lines of Figure 5, practically coincide. This can be explained considering that we are interested in the estimation of relaxation times, i.e., of decay rates. In order to achieve such a goal, it is crucial that the measurement of the signal decrease, i.e., the ratio of the signal acquired in two different echo times. Therefore, instead of values T_{E}, it is only important the difference between them. A third echo time, T_{E}(3), equal to T_{E}(2), allows us to compute twice the signal decay, which is the quantity in which we are interested;
the red and the green lines of Figure 5 show a clear trend: their values grow linearly when increasing T_{2}. In particular, we found that the distance with the blue line (i.e., lowest T_{E}, 82 ms) is a little bit bigger than the value of the considered spin-spin relaxation time, T_{2}. For example, in the case of T_{2} = 100 ms, the ideal echo times were T_{E} = [83, 197, 205] ms; the last two values are approximately 110% of (T_{E}(1) + T_{2}). By considering the other simulated T_{2} values, we found that this coefficient is 110% ± 10%. Within this range, the CRLB of T_{2} can be considered constant.
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) + T_{2}), considering the T_{2} of the tissue in which we are interested.
A similar evaluation has been conducted for the minimization of T_{1} 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_{R}s have similar values, which, starting from 500 ms in the case of tissue with T_{1} = 700 ms, grow almost linearly up to 1, 400 ms for tissues with higher T_{1} (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 < T_{1} < 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 T_{1}, as far as T_{R} from T_{2}, 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 T_{2}, but very different T_{1} values share the same three optimal echo times for T_{2} 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 T_{1} and T_{2}, 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 T_{2} 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.
Numerical Experiments
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 T_{1}T_{2}] = [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, T_{1} and T_{2} 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 T_{2} 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, T_{1} and T_{2} 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 T_{1}T_{2}] = [5 700 68] (spinal cord) and [ρ T_{1}T_{2}] = [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 T_{1} and T_{2} 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 T_{1} estimator is not pronounced. Once again, the result highlights the need of properly tuning the acquisition parameters.
Conclusions
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 T_{1} and T_{2} 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 T_{1} and T_{2}) 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.
Conflict of Interest
The authors declare no conflict of interest.
Appendix
From [16], CRLBs may also be expressed in a slightly different form with respect to Equation (6). In particular, it yields:
−E[∂2lnp(y;θ)∂θ2]=E[(∂lnp(y;θ)∂θ)2]From Equation (4), the log-likelihood function related to N complex acquisitions is:
log[p(y;θ)]=−Nlog(2πσ2)−12σ2∑k=1N[f2(θ)+yR2(k)+yI2(k)−2f(θ)yR(k)cos(ϕ)−2f(θ)yI(k)sin(ϕ)]where subscript k refers to k-th acquisition, i.e., the MRI scan with parameters T_{R}(k), T_{E}(k). The partial derivatives can be computed as:
∂lnp(y;θ)∂ρ=−12σ2∑k=1N[∂f2(θ)∂ρ−2∂f(θ)∂ρ(yR(k)cos(ϕ)+yI(k)sin(ϕ))]∂lnp(y;θ)∂T1=−12σ2∑k=1N[∂f2(θ)∂T1−2∂f(θ)∂T1(yR(k)cos(ϕ)+yI(k)sin(ϕ))]∂lnp(y;θ)∂T2=−12σ2∑k=1N[∂f2(θ)∂T2−2∂f(θ)∂T2(yR(k)cos(ϕ)+yI(k)sin(ϕ))]where the first order derivatives are:
∂f(θ)∂ρ=exp(−TET2)[1−exp(TRT1)]∂f2(θ)∂ρ=2ρexp(−2TET2)[1−exp(TRT1)]2∂f(θ)∂T1=−ρTRT12exp(−TET2)exp(−TRT1)∂f2(θ)∂T2=−2ρ2TRT12exp(−2TET2)exp(−TRT1)[1−exp(−TRT1)]∂f(θ)∂T2=ρTET22exp(−TET2)[1−exp(TRT1)]∂f2(θ)∂T2=2ρ2TET22exp(−2TET2)[1−exp(TRT1)]2
In order to compute the expected value of Equation (7), Monte Carlo simulations have to be implemented.
ReferencesSlichterP.ChoZ.H.JonesJ.SinghM.HaleyA.P.Knight-ScottJ.FuchsK.L.SimnadV.I.ManningC.Shortening of hippocampal spin-spin relaxation time in probable Alzheimer's disease: A 1H magnetic resonance spectroscopy studyAntoniniA.LeendersK.L.MeierD.OertelW.H.BoesigerP.AnlikerM.T2 relaxation time in patients with Parkinson's diseaseMariappanS.V.S.SubramanianS.ChandrakumarN.RajalakshmiK.R.SukumaranS.S.Proton relaxation times in cancer diagnosisRoebuckJ.R.HakerS.J.MitsourasD.RybickiF.J.TempanyC.M.MulkernR.V.Carr-Purcell-Meiboom-Gill imaging of prostate cancer: Quantitative T2 values for cancer discriminationDrainL.E.A direct method of measuring nuclear spin-lattice relaxation timesBarralJ.K.GudmundsonE.StikovN.Etezadi-AmoliM.StoicaP.NishimuraD.G.A robust methodology for in vivo T1 mappingChangL.C.KoayC.G.BasserP.J.PierpaoliC.Linear least-squares method for unbiased estimation of T1 from SPGR signalsTrzaskoJ.D.MostardiP.M.RiedererS.J.ManducaA.Estimating T1 from multichannel variable flip angle SPGR SequencesBonnyJ.M.ZancaM.BoireJ.Y.VeyreA.T2 maximum likelihood estimation from multiple spin-echo amplitude imagesSijbersJ.den DekkerA.J.RamanE.van DyckD.Parameter estimation from Magnitude MR imagesLiuJ.NieminenA.O.KoenigJ.L.Calculation of T1 T2 and proton spin density in nuclear magnetic resonance imagingFennessyF.M.FedorovA.GuptaS.N.SchmidtE.J.TempanyCM.MulkernR.V.Practical considerations in T1 mapping of prostate for dynamic contrast enhancement pharmacokinetic analysesJonesA.J.HodgkinsonP.BarkerA.L.HoreP.J.Optimal Sampling Strategies for the Measurement of Spin-Spin Relaxation TimesKayS.M.GudbjartssonH.PatzS.The Rician distribution of noisy MRI dataBaseliceF.FerraioliG.PascazioV.Relaxation time estimation from complex magnetic resonance imagesSpandonisY.HeeseF.P.HallD.H.High resolution MRI relaxation measurements of water in the articular cartilage of the meniscectomized rat knee at 4.7TAnastasiouA.HallL.D.Optimisation of T_{2} and M_{0} measurements of bi-exponential systemsWrightG.A.Magnetic resonance imagingWangY.LeiT.Statistical Analysis of MR Imaging and Its Applications in Image ModelingProceedings of IEEE International Conference Image ProcessingAustin, TX, USA13–16 November 1994866870BaseliceF.FerraioliG.ShabouA.Field map reconstruction in magnetic resonance imaging using Bayesian estimationEggersH.KnoppT.PottsD.Field inhomogeneity correction based on gridding reconstruction for magnetic resonance imagingStaniszG.J.OdrobinaE.E.PunJ.EscaravageM.GrahamS.J.BronskillM.J.HenkelmanR.M.T1, T2 relaxation and magnetization transfer in tissue at 3TFigures and Tables
Square root of the Cramer–Rao lower bound (CRLB) for proton density (blue), spin-lattice (T_{1}) relaxation time (green) and spin-spin (T_{2}) relaxation time (red) for different signal-to-noise ratio values expressed in decibels (logarithmic scale). CRLB values have been normalized for the parameter values in order to be plotted in the same graph.
The square root of the CRLB of proton density ρ (a), T_{1} relaxation time (b) and T_{2} relaxation time (c) for different numbers of acquisitions.
The square root of the CRLB of proton density ρ (a), T_{1} relaxation time (b) and T_{2} relaxation time (c) for different combinations of T_{R} and T_{E} values in the case of ρ = 2.5, T_{1} = 1,600 ms and T_{2} = 90 ms.
The square root of the CRLB of proton density ρ (a), T_{1} relaxation time (b) and T_{2} relaxation time (c) for different combinations of T_{R} and T_{E} values in the case of ρ = 2.8, T_{1} = 1,800 ms and T_{2} = 60 ms.
T_{E} values that minimize the CRLB of T_{2} for tissues with different spin-spin relaxation times, T_{2}. Three values have been considered: the blue line is for the lowest T_{E} value, the red line for the highest one and green for the intermediate one.
T_{R} values that minimize the CRLB of T_{1} for tissues with different spin-lattice relaxation times, T_{1}. Three values have been considered: the red line is for the highest T_{R} value, the blue line for the lowest one and green for the intermediate one.
Optimal T_{E}(2) (a) and T_{R}(2) (b) values considering nine acquisitions in the case of tissues with different T_{1} and T_{2} relaxation times. It can be noticed that the T_{E}(2) value is substantially independent from tissue spin-lattice relaxation time T_{1}, as far as T_{R}(2) from spin-spin relaxation time T_{2}.
Square root of the CRLB for proton density (blue), spin-lattice (T_{1}) relaxation time (green) and spin-spin (T_{2}) relaxation time (red) for the dataset with different acquisition parameters. CRLBs values have been normalized for the parameter values in order to be plotted in the same graph.
The empirical probability density function of the ρ (a), T_{1} (b) and T_{2} (c) estimators in the case of Dataset 1 (blue line), Dataset 2 (green line) and Dataset 3 (red line). The true values are ρ = 5.5, T_{1} = 775 and T_{2} = 44.5, respectively.
The empirical probability density function of the T_{1} (a) and T_{2} (b) estimators in the case of Dataset 1 (blue line) and Dataset 2 (red line). Dataset 1 (blue line) imaging parameters are ideal for tissues with lower T_{1} and T_{2}. On the contrary, Dataset 2 (red line) is ideal for the tissue with higher relaxation times.
Optimal repetition times, T_{R}, for T_{1} estimation in case of different tissues and numbers of acquisitions at 1.5 T and 3 T.
Tissue
1.5T
3T
2 images
3 images
4 images
2 images
3 images
4 images
liver
490; 4000
490; 510; 4,000
380; 490; 510; 4,000
650; 4,000
570; 650; 4,000
570; 570; 650; 4,000
skeletal muscle
720; 4,000
720; 720; 4,000
680; 720; 720; 4,000
1,090; 4,000
990; 1,090; 4,000
870; 990; 1,090; 4,000
heart
840; 4,000
770; 840; 4,000
770; 790; 840; 4,000
1,060; 4,000
910; 1,060; 4,000
770; 910; 1,060; 4,000
kidney
570; 4,000
430; 570; 4,000
430; 460; 570; 4,000
910; 4,000
790; 910; 4,000
750; 790; 910; 4,000
cartilage
760; 4,000
690; 760; 4,000
630; 690; 760; 4,000
880; 4,000
770; 880; 4,000
770; 780; 880; 4,000
white matter
690; 4,000
690; 710; 4,000
640; 690; 710; 4,000
850; 4,000
780; 850; 4,000
730; 780; 850; 4,000
gray matter
920; 4000
840; 920; 4000
800; 840; 920; 4,000
1,150; 4,000
980; 1,150; 4,000
910; 980; 1,150; 4,000
optic nerve
960; 4,000
960; 1,060; 4,000
960; 1,060; 1,100; 4,000
970; 4,000
910; 970; 4,000
910; 970; 1,030; 4,000
spinal cord
600; 4,000
550; 600; 4,000
450; 550; 600; 4,000
760; 4,000
660; 760; 4,000
660; 700; 760; 4,000
blood
1,120; 4,000
840; 1,120; 4,000
830; 840; 1,120; 4,000
1,120; 4,000
1,040; 1,120; 4,000
1,030; 1,040; 1,120; 4,000
Optimal echo times T_{E} for T_{2} estimation in case of different tissues and numbers of acquisitions at 1.5 T and 3 T.
Tissue
1.5T
3T
2 images
3 images
4 images
2 images
3 images
4 images
liver
82; 134
82; 134; 138
82; 134; 138; 146
82; 134
82; 134; 134
82; 134; 134; 142
skeletal muscle
82; 130
82; 130; 138
82; 130; 138; 1;400
82; 132
82; 132; 144
82; 132; 144; 146
heart
82; 124
82; 124; 134
82; 124; 134; 136
82; 132
82; 132; 140
82; 132; 140; 148
kidney
82; 158
82; 158; 168
82; 158; 168; 188
82; 144
82; 144; 154
82; 144; 154; 164
cartilage
82; 116
82; 116; 116
82; 116; 116; 122
82; 114
82; 112; 114
82; 112; 114; 120
white matter
82; 162
82; 162; 188
82; 162; 188; 208
82; 162
82; 162; 178
82; 162; 178; 214
gray matter
82; 210
82; 210; 244
82; 210; 244; 280
82; 192
82; 192; 218
82; 192; 218; 240
optic nerve
82; 192
82; 192; 222
82; 192; 222; 250
82; 168
82; 168; 196
82; 168; 196; 240
spinal cord
82; 160
82; 160; 192
82; 160; 192; 208
82; 174
82; 174; 190
82; 174; 190; 212
blood
82; 516
82; 516; 558
82; 516; 558; 620
82; 436
82; 436; 562
82; 436; 562; 588
Mean spin-lattice and spin-spin relaxation times for the considered tissues at 1.5 T and 3 T.
Tissue
1.5T
3T
T_{1}
T_{2}
T_{1}
T_{2}
liver
576
46
818
42
skeletal muscle
1,008
44
1,412
50
heart
1,030
40
1,471
47
kidney
690
55
1,194
56
cartilage
1,038
44
1,156
43
white matter
884
72
1,084
69
gray matter
1,124
95
1,820
99
optic nerve
815
77
1,083
78
spinal cord
745
74
993
78
blood
1,441
290
1,932
275
Optimal echo times for T_{2} estimation of gray matter for acquisitions at 1.5 T in the case of different minimum T_{E} values.
2 images
3 images
4 images
minimum T_{E} = 82 ms
82, 210
82, 210, 244
82, 210, 244, 280
minimum T_{E} = 50 ms
50, 182
50, 182, 212
50, 182, 212, 234
minimum T_{E} = 20 ms
20, 158
20, 158, 180
20, 158, 180, 210
CRLBs for equally and optimally spaced T_{R} and T_{E} values.
Tissue parameter
CRLB: 16 images Equispaced
CRLB: 16 images Optimized
Improvement (%)
CRLB: 12 images Optimized
Improvement (%)
ρ
0.1562
0.1291
17.34%
0.1506
3.58%
T_{1}
3144
1483
52.83%
1.960
37.66%
T_{2}
2.708
1.808
33.23%
2.144
20.83%
Acquisition parameters: three datasets composed of four images.
Repetition Times (s)
Echo Times (ms)
SNR (dB)
Dataset 1
0.55, 4.0
80, 140
30
Dataset 2
0.75, 4.0
80, 170
30
Dataset 3
0.90, 4.0
80, 200
30
Estimator performances: three datasets composed of four images.
Parameter
True Value
Dataset 1
Dataset 2
Dataset 3
Mean
Variance
Mean
Variance
Mean
Variance
ρ̂
5.5
5.52
0.20
5.58
0.47
5.79
2.56
T̂_{1}
775
776.1
1662
776.5
2615
776.5
3, 566
T̂_{2}
44.5
44.54
3.58
44.49
7.95
44.32
20.82
Acquisition parameters in the case of two tissues; the datasets are composed of four images.