Optimal Echo Spacing for Multi-Echo Imaging measurements of Bi-exponential T₂ relaxation

Abstract

Calculations, analytical solutions, and simulations were used to investigate the trade-off of echo spacing and receiver bandwidth for the characterization of bi-exponential transverse relaxation using a multi-echo imaging pulse sequence. The Cramer-Rao lower bound of the standard deviation of the four parameters of a two-pool model were computed for a wide range of component T₂ values and echo spacing. The results demonstrate that optimal echo spacing (TE_opt) is not generally the minimal available given other pulse sequence constraints. The TE_opt increases with increasing value of the short T₂ time constant and decreases as the ratio of the long and short time constant decreases. A simple model of TE_opt as a function of the two T₂ time constants and four empirically derived scalars is presented.

INTRODUCTION

Characterizing transverse relaxation with multiple exponentials, or a T₂ spectrum, is of interest in the study of various tissues and samples, including white matter and nerve (1,2), skeletal muscle (3,4), cerebral injury (5), tumor (6), plants (7), food (8), bone marrow (9,10) and more. Such characterization offers the potential to indirectly observe microscopic sample characteristics, such as myelin in white matter; however, it requires a relatively high signal-to-noise ratio (SNR) (11–13), so optimization of acquisition and processing methods is important.

Some studies have investigated optimal sampling of multi-exponential transverse relaxation, including numerical evaluations of number and range of echo times sampled (12–14) and the benefits of log-spaced sampling (15). Beyond NMR-specific studies, there are a wealth of publications that address the fitting and parameter estimation from models involving the sum of exponential functions. An extensive review of this material was presented by Istratov and Vyvenko (16) and includes discussion on T₂-spectral resolution limits based on sampling times and model parameters. More recently, at least one publication presented simple analytical approximations of the uncertainty of fitted parameters in models of bi-exponential relaxation (17), which could be used to optimize sampling. A couple of studies have incorporated practical imaging factors/limitations –the effect of imperfect refocusing (18) and SAR limitations on sampling times (19) – into the process of optimizing sampling, but no study, to our knowledge, has incorporated the effect of receiver bandwidth on sampling times.

In a multi-echo imaging measurement, the time required for sampling each echo is typically on the order of milliseconds and can often place a lower limit on the echo spacing (TE). In order to reduce this time and, in-turn, TE, one must increase the receiver bandwidth (BW) at the cost of SNR. Herein is presented calculations and simulations that demonstrate the effect of trading-off BW for TE for two-pool systems with a range of possible relaxation rates and pool sizes.

THEORY

As shown in Fig. 1, the minimum TE of a multi-echo imaging pulse sequence suitable for measuring transverse relaxation depends on the time required to acquire each echo (T_acq) and the time required for fixed events (T_const), like the RF refocusing pulse, spoiler gradients, ramp-time delays and delays for eddy current decay:

$$TE=T_{\mathit{const}}+T_{\mathit{acq}}=T_{\mathit{const}}+N_s/BW,$$ [1]

where, N_s is the number of complex samples collected during each echo and BW is the bandwidth of this acquisition (assuming quadrature detection). Generally, every effort is made to minimize both T_acq and the T_const in order to minimize TE. Assuming one has defined a minimum N_s based on image resolution requirements, the only way to further reduce T_acq is to increase BW. The inherent trade-off in this reduction of T_acq is that the image SNR is inversely proportional to the square root of BW, so reducing TE through increased BW also decreases the SNR. A reduced TE will improve the precision of estimated transverse relaxation parameters, while the concomitantly increased BW and subsequently decreased SNR will deteriorate this precision. Note that the exception to this description is when the total number of echoes (NE) is limited and the sample is comprised of a combination of spins with short and long T₂. In this case, reducing TE may reduce precision of fitted parameters as a consequence of under-sampling of long T₂ components. This problem may be avoided with minimal complication by appending a small number of widely spaced echoes at the end of the standard echo train (19). For the purpose of the work herein, this under-sampling of long T₂ signal is avoided by assuming there is no practical limit on NE, so the analysis is focused on the trade-off of TE for image SNR.

Fig 1.

Relevant timings of a multi-echo imaging pulse sequence. Two consecutive echoes are shown, separted by TE, one RF refocusing pulse, and two spoiler gradients.

In order to determine the optimal TE for a given system, the Cramer-Rao lower bound (CRLB) of the estimated relaxation parameter variance can be computed. A complete derivation of the Cramer-Rao lower bound can be found elsewhere (20), but a simple explanation in the context of this paper is as follows. Consider a two-pool system, in which the observed transverse magnetization is described by a simple real bi-exponential function with added white Gaussian noise,

$$M_T(n)=M_o^{a}exp(-n·TE/T_2^a)+M_o^{b}exp(-n·TE/T_2^b)+\epsilon (n),$$ [2]

where n = [1,2,…, NE], and ε(n) are independent and identically distributed random values drawn from a Gaussian distribution with zero mean and standard deviation (SD) σ (i.e., the image noise). Unbiased estimates of the four model parameters, $M_o^a,M_o^b,T_2^a$, and $T_2^b$, result from a least-squares fitting of the model to a series of observations, M_T(n). Then the CRLB of the SD of the k^th of these fitted parameters, s(θ_k), is defined by

$$s({\theta }_k)=\sqrt{{({\mathbf{F}}^{-1})}_{kk}},$$ [3]

where θ_k represents the k ^th fitted parameter and F is the Fisher information matrix given by

$$F_{jk}=\frac{1}{{\sigma }^2}\sum _n\left(\frac{\partial M_T(n)}{\partial {\theta }_j}\frac{\partial M_T(n)}{\partial {\theta }_k}\right).$$ [4]

That is, the unbiased estimate parameter θ_k has an associated variance that is no less than s²(θ_k), thereby defining the best precision possible in estimating each of the four model parameters, $M_o^a,M_o^b,T_2^a$, and $T_2^b$. The elements inside the summation in Eq [4] are easily defined algebraically for the bi-exponential system in Eq [2], and, using Eq [1], TE and σ are related by,

$$\sigma ={\sigma }_0\sqrt{BW/B{W}_0}={\sigma }_0\sqrt{\frac{N_s}{(TE-T_{\mathit{const}})B{W}_0}},$$ [5]

where σ₀ defines the SD of the image noise at a receiver bandwidth of BW₀. From here, both numerical and analytical solutions of Eq [3]–[5] are possible, as are Monte Carlo simulations of fitting Eq [2]. All of these methods provide s(θ_k) as a function of the four sample parameters ( $M_o^a,M_o^b,T_2^a$, and $T_2^b$), and the acquisition parameters (NE, TE, T_const, N_s, BW₀, and σ₀), although the CRLB solutions are closed form and much faster than using the Monte Carlo approach.

This work focuses on primarily on the numerical solutions, because of their ease and efficiency, but some Monte Carlo solutions and analytical solutions are presented for validation and generality. Simulations were also run to investigate the effect of TE when the underlying model is more complex than a sum of two discrete exponential functions. For example, in practice, most investigators characterizing transverse relaxation in white matter assume that the two commonly observed T₂ components have a finite width in T₂-space. Consequently, fitting of these data is usually done using a linear-inverse approach, where the observed signal is fitted to a wide range of decaying exponential functions and the solution (the T₂-spectrum) is regularized by minimizing its energy or curvature (11).

METHODS

Numerical Solutions

As a starting point for numerical calculations, values of T_const = 5 ms, NE = 200, σ₀ = 1/750, Ns = 128, BW₀= 64 kHz, $M_o^a=0.2$ and $M_o^b=0.8$ were used. This T_const value was based on a 1 ms RF refocusing pulse, two 1-ms spoiler gradients and two 1-ms delays after the spoiler gradients to allow eddy-currents to decay. (For more information on imaging pulse sequence requirements for measuring multi-exponential T₂, see (21) and (22), and related literature.) The effect of variations in T_const on the CRLB calculations is discussed further below. The relatively large NE value ensured that varying TE did not result in under-sampling of the longer-lived T₂ signal decay, which was tested by re-running a subset of CRLB calculations without using Eq [5] to incorporate the effects of minimum TE on image noise (i.e., σ is constant) – see Fig 2, in Results section. (Note that while NE = 200 may not be practical for most in-vivo imaging for reasons of power deposition and spoiler gradient demands, sampling to long echo times by appending a small number of widely spaced echoes at the end of a standard multi-echo sequence is practical and has the same effect of preventing under-sampling of long T₂ signal (19)). The values of the remaining acquisition parameters – σ₀, BW₀, Ns, $M_o^a$, and $M_o^b$ – are arbitrary as they do not influence the optimization of TE v. BW – also demonstrated in Fig 2, in the Results section.

Fig 2.

Plots of SNR of four fitted parameters (indicated by line color) of a bi-exponential model of transverse relaxation as a function of TE. The zenith of each plot is indicated with a diamond symbol. In frame a), calculations were made using $T_2^a=15\text{ms},T_2^b=75\text{ms}$, T_const = 5 ms, NE = 200, σ₀ = 1/750, Ns = 128, BW₀ = 64 kHz, $M_o^a=0.2$ and $M_o^b=0.8$. Other frames show results from the same calculations made using the following different parameters: b) $M_o^a=0.4$ and $M_o^b=0.6$, c) σ₀ = 1/250, and d) N_s = 256. Also shown in frame a) as dashed lines are the results from the same calculations make without the use of Eq [5] – i.e, no dependence of σ on TE.

With these parameters fixed, a series of calculations were performed with varied $T_2^a,T_2^b$ and TE. The short-lived T₂ component was varied linearly as $T_2^a=[10,11,\dots ,35]\text{ms}$, which spanned the expected range of T₂s for myelin water and intra-cellular muscle water. The long-lived relaxation time was varied in linear proportion to the short-lived time as $T_2^b=T_x·T_2^a$, where T_x =[3,3.25,3.5,···,10], which was more than sufficient to span the range of expected long-lived T₂ components in white matter, nerve and muscle. For each pair of $T_2^a$ and $T_2^b$, the Fisher Information matrix and resultant CRLB for estimated parameters’ SD were computed using Eq [3] – [5] for TE=[5.5,5.6,5.7,···,40] ms. The SNR of each fitted parameter was then defined as

$$\mathit{SNR}({\theta }_k)={\theta }_k/s({\theta }_k).$$ [6]

Simulations

In order to validate these CRLB calculations and to explore a wider range of possible systems, a series of Monte Carlo simulations were performed. To validate the CRLB calculations, noisy bi-exponential relaxation data were generated then fitted with Eq [2] using a Levenberg-Marquardt algorithm. The initial guesses for the regression were randomly varied for each trial, with means equal to the underlying model parameters and a 10 % coefficient of variation. These simulations used the following parameters: NE = 200, σ₀ = 1/750, Ns = 128, BW₀ = 64 kHz, $M_o^a=0.2$ and $M_o^b=0.8,T_2^a=[10,15,20,25]\text{ms}$, T_x =[3,4,6,8,10], and TE=[5.5,6.0,6.5,···,30] ms. Zero mean Gaussian noise (ε(n), n = 1,…, NE), with SD as defined by Eq [5] were independently generated for N_t = 1000 trials, using each combination of $T_2^a$, T_x and TE. The SNR for each parameter was then defined as the ratio of the parameter value to the SD of its fitted value calculated across the N_t trials, similar to Eq [6].

Simulations were also run to investigate the effect of TE on model systems comprised of a distribution of relaxation times rather than two distinct components. In particular, the model described above was modified such that each spin pool was defined by a Gaussian shape in a log-spaced T₂ domain (similar to used in a previous study for fitting relaxation data (23)). That is, component a was defined by

$$S^a(j)=p^{a}exp\left(-{\left(\frac{log{T}_2(j)-log{T}_2^a}{logd}\right)}^2\right),$$ [7]

(where log is the natural logarithm) and likewise for S^b(j), where d determines the width of the distribution, and p^a and p^b are set such that the sum of S^a(j)and S^b(j)over all j = 1 to J equaled $M_o^a$ and $M_o^b$, respectively. For all simulations, T₂(j)was defined by J = 100 values, log-spaced between 5 ms and 1 s. With these distributions, the observed signal was then defined as

$$M_T(n)=\sum _{j=1}^J\left[(S^a(j)+S^b(j))exp(-n·TE/T_2(j))\right]+\epsilon (n).$$ [8]

The simulations used $T_2^a=15\text{ms}$, T_x =[3,4,6,10], d=[1.0, 1.26, 1.59, 2.0], and 13 TE values pseudo-log spaced between 5.5 ms and 30 ms. (Note that for the cases where d = 1.0, the T₂ component width was infinitely narrow and Eq [2] was used to create M_T(n)). Figure 6 shows the T₂ spectra (sum of S^a(k)and S^b(k)) for each $T_2^a$, T_x and d. All other parameters were the same as for the bi-exponential relaxation simulations, defined above. For every combination of T_x and d, noise, ε(n), was independently generated for 1000 trials.

Fig 6.

T₂ spectra, defined using Eq [7], used for Monte Carlo simulations of fitting data comprised of distributions of T₂ times. Shown in the four frames are spectra defined by $T_2^a=15\text{ms}$ (all cases), four values of $T_2^b$ and four values of component width (d).

Each simulated noisy signal generated by Eq [8] was fitted to a range of 100 T₂ values, log-spaced between 5 ms and 1 s using a non-negative least square method (24) and regularized with a minimum curvature constraint (11). The regularizing parameter was automatically adjusted using the generalized cross-validation approach (25). Each spectrum was then analyzed by decomposing it into n + 1 T₂ components, where n was the number of spectral nadirs identified by positive to negative changes in the first derivative of the spectrum. After discarding T₂ components representing < 2% of the integrated spectral amplitude, if exactly two T₂ components were identified, then four model parameters, $M_o^a,M_o^b,T_2^a$, and $T_2^b$, were computed. Component amplitudes, $M_o^a$ and $M_o^b$, were defined as the integrated area of each T₂ component and the component T₂ values, $T_2^a$ and $T_2^b$, were defined as the amplitude-weighted mean T₂ value computed over each component T₂ domain.

Analytical Solutions

The appendix outlines a general analytical solution for the standard deviation of each fitted parameter. The only approximation involved was to assume NE = ∞. This is equivalent to requiring that the decay of transverse magnetization be sampled down to the noise floor to avoid under-sampling of the long lived T₂ component, as described above for the numerical calculations.

RESULTS AND DISCUSSION

Figure 2a demonstrates typical CRLB-calculated graphs of SNR(θ_k) v. TE for each of the four fitted parameters for a system defined by $T_2^a=15\text{ms},T_2^b=75\text{ms},M_o^a=0.2$ and $M_o^b=0.8$. The solid lines are SNR(θ_k) values calculated using Eq [3]–[6], while the dashed lines are derived from the same calculation made while excluding the influence of BW on image noise (i.e, without Eq [5]). The dashed lines decrease monotonically with TE, which agrees with previous work (14) (which used CRLB, but did not incorporate a BW-TE relationship) and demonstrates that under-sampling of long T₂ components was not a significant factor in the results presented herein. In contrast, for each fitted parameter, the solid lines show that SNR(θ_k) increases with TE to some maximal value, denoted in the figure by a diamond symbol, then decreases monotonically with further increasing TE. This demonstrates that the influence of echo spacing on BW and, in turn, image noise, is an important factor in characterizing multi-exponential relaxation.

Also shown in Fig 2 are similar graphs made from three variations in the sample or acquisition parameters: b) $M_o^a=0.4$ and $M_o^b=0.6$, c) σ₀ = 1/250, and d) N_s = 256. In all cases, the optimal TE (TE_opt) for all four estimated parameters are identical to those in frame a), demonstrating that the TE_opt calculations are independent of compartment sizes, baseline SNR, and number of samples. This independence from $M_o^a,M_o^b$, σ₀, and N_s can also be seen in the analytical solutions presented in the appendix, when combined with Eq [5] and [6]. For example, Eq A.3 shows CRLB-defined minimum variance of all four model parameters. In each case, the parameters $M_o^a,M_o^b$, σ₀, and N_s are either not present or can be factored out. Therefore, each of these four parameters may change the scale of s(θ), but not the shape of its dependence on TE.

In addition to the numerical and analytical solutions, Monte Carlo simulations were also performed. Figure 3 shows plots of $\mathit{SNR}(M_o^a)\text{v}$. TE, derived from numerical CRLB calculations (lines) and the Monte Carlo simulations (dots) for the bi-exponential model given by Eq. [2]. The results from the analytical solutions are not shown but would be indistinguishable from the numerical calculations. With the exception of a few measurements with low $\mathit{SNR}(M_o^a)$, the Monte Carlo- and CRLB-derived calculations are in good agreement, thereby validating the CRLB calculations and analytical solutions. In the cases where the Monte Carlo derived measures of $\mathit{SNR}(M_o^a)$ do not reach those determined from the CRLB (e.g., around TE = 18 ms in Fig 3a), the difference likely results from very low SNR and, as a consequence, ineffective convergence to the true least-square solution in these cases.

Fig 3.

Plots of $\mathit{SNR}(M_o^a)$ as a function of TE for a wide range of different $T_2^a$ and $T_2^b$ values. It is apparent that TE_opt increases with increases $T_2^a$ and with decreasing $T_2^b$.

Figure 3 also demonstrates the strong dependence of TE_opt on both $T_2^a$ and $T_2^b$. This is demonstrated by the solid line curves in Fig 3, which show $\mathit{SNR}(M_o^a)$ for a wide array of different $T_2^a$ and $T_2^b$ values. Comparing data across the four frames shows that TE_opt increases with increasing $T_2^a$, while comparing data within each frame shows that TE_opt increases with decreasing $T_2^b$. The increase in TE_opt with increasing $T_2^a$ is not surprising and simply indicates that a more slowly decaying function need not be sampled as quickly as a more quickly decaying function to produce the same variance of estimated parameters. The increasing TE_opt with decreasing $T_2^b$ is, perhaps, less intuitive, and can be interpreted that SNR becomes increasingly more valuable as compared to temporal sampling density (i.e., echo spacing) when trying to distinguish signal components with increasingly similar T₂s.

These CRLB calculations are relatively easy to compute for any given system of bi-exponential relaxation, but for a quick reference, the data from the calculations presented herein were used to generate a simple empirical model of the relationship between TE_opt and $T_2^a$ and $T_2^b$. Figure 4 shows a family of curves plotting TE_opt v. $T_2^a$ for all ratios $T_2^a/T_2^b$, from which it was observed that TE_opt increases approximately linearly with $T_2^a$:

$${TE}_{\mathit{opt}}=m_1{T}_2^a+b_1,$$ [9]

and the slope (m₁) and intercept (b₁) of these linear functions vary with $T_2^a/T_2^b$. Figure 5 shows a crudely linear relationships between log(m₁) and $T_2^a/T_2^b$ and between b₁ and $T_2^a/T_2^b$, and from these, Eq. [9] can be expanded to

Fig 4.

A family of curves showing TE_opt vs $T_2^a$ for a wide range of $T_2^a/T_2^b$ ratios. Each solid line is a best fitted linear function for a given $T_2^a/T_2^b$ ratio.

Fig 5.

(left) Plot of the natural logarithm of the slopes of the curves in Fig 4 vs $T_2^a/T_2^b$. The solid line shows the best fit linear function to these data as described by the equation in the frame. (right) Plot of the intercepts of the curves in Fig 4 vs $T_2^a/T_2^b$. The solid line shows the best fit linear function to these data as described by the equation in the frame.

$${TE}_{\mathit{opt}}\approx a_1{T}_2^{a}exp\left(m_2\frac{T_2^a}{T_2^b}\right)+m_3\left(\frac{T_2^a}{T_2^b}\right)+b_3,$$ [10]

where a₁= exp(b₂). Thus, three linear regressions were calculated to produce estimates of m₁, m₂, m₃, b₁, b₂, and b₃, resulting in the four independent constants in Eq [10]: a₁, m₂, m₃, and b₃. The same approach was used for all four estimated parameters in Eq [1] ( $M_o^a,M_o^b,T_2^a$, and $T_2^b$), and the results are shown in Table 1. Equation [10] thus provides a quick and simple formula to estimate TE_opt for a given two-pool system and for a given parameter of interest.

Table 1.

Constants computed for optimizing TE_opt with Eq. [8]

Parameter of interest	Constant in Eq. [8]
	a1	m2	m3 (s)	b3 (s)
$M_o^a$	0.072	8.03	−0.020	0.0078
$M_o^b$	0.55	−1.98	−0.0086	0.0094
$T_2^a$	0.22	−0.11	−0.0009	0.0063
$T_2^b$	0.088	−2.35	−0.013	0.011

In addition to the numerical studies, a complete analytical solution for s(θ_k) in the bi-exponential model is presented in the appendix. As mentioned above, the results match the numerical solutions and have the advantage of being applicable to arbitrary conditions, beyond those explored in this paper. These analytical results also provide insight into signal dependencies that are not readily apparent from the numerical results. For example, while the numerical results presented demonstrate that $s(M_o^a)$ increases as $T_2^a$ approaches $T_2^b$, the analytical solution of $s(M_o^a)$ shows this effect quantitatively with the ${(e^{-TE/T_2^a}-e^{-TE/T_2^b})}^3$ term in the denominator. Similarly, one can see that the effect of similar T₂s is more pronounced for estimating component amplitudes than time constants.

In comparison to the analytical solutions presented herein, much simpler, although approximate, solutions have been derived using Bayesian probability theory (17). These equations produce qualitatively similar curves to the dashed lines in Fig 2, but when combined with Eq [5] do not predict the existence of an optimal TE as found with the CRLB approach and validated with Monte Carlo, herein. Thus, while the CRLB solutions are complex analytically, they ultimately provide a more complete picture of the effect of model and acquisition parameters on estimated parameter variance.

A potential shortcoming of the CRLB solutions lies in the fact that a strict bi-exponential model, as defined by Eq [2], is probably not a good representation of multi-exponential relaxation in many tissues and samples. A more relevant model is presented in Eq [8], which generalizes the bi-exponential model to one defined by two smooth distributions of relaxation times, as shown in Fig 6. Figure 7 shows the results of the Monte-Carlo simulations of fitting data generated using these smooth T₂ spectra. (Note that these results were derived only from trials that resulted in two fitted T₂ components, which was > 88% of trials for all but three cases shown: d = 1.59, $T_2^a=45\text{ms}$, and d = 2.00, $T_2^a=45$, and 60 ms.) The results demonstrate that, as expected, the CRLB calculations presented above do not predict the absolute value of the estimated parameter variances but they do predict the general shape and model-parameter dependence of $\mathit{SNR}(M_o^a)\text{v}$. TE. Note similarity between Fig 7a with Fig 3b, which shows the results of fitting the same underlying bi-exponential data with a strict bi-exponential model (Fig. 3b) and with a distribution of T₂ times (Fig 7a). Naturally, the strict bi-exponential fitting results in slightly higher $\mathit{SNR}(M_o^a)$ values, particularly at lower values of $T_2^a/T_2^b$, but the $\mathit{SNR}(M_o^a)\text{v}$. TE curve shape and TE_opt values are similar. Also, as the model T₂ components are broadened (increasing values of d, Fig 7b–d), $\mathit{SNR}(M_o^a)$ values drop and the $\mathit{SNR}(M_o^a)\text{v}$. TE curve broadens but the TE_opt values do not change appreciably.

Fig 7.

Plots of $\mathit{SNR}(M_o^a)$ as a function of TE for cases where a distribution of T₂ times were fitted with a distribution of decaying exponential functions. Results are shown for a range of different $T_2^b$ and component width (d) values.

A more general interpretation of the statistics of fitting smooth T₂ spectra is a complicated problem involving many factors. In addition to breadth of the T₂ components, as considered herein, the number and range of exponential functions to fit, the method of regularization, the adjustment of the regularizing parameter, and the method extracting model parameters from the spectrum may all significantly impact the results. Nonetheless, Eq [10] appears to provide a good starting point for estimating TE_opt in systems that are thought to be well described by two relaxation components.

The utility of this work, through either the numerical or analytical solutions, is possibly most significant for myelin water mapping in white matter (26,22,27). For these studies, a two pool model is often used to describe water from within the layers of myelin as pool a and water from the intra- and extra-axonal spaces lumped together as pool b, and the relevant parameter for optimization is $\mathit{SNR}(M_o^a)$ because the myelin content is believed to be proportional to by $M_o^a$. In-vivo at 1.5T, the commonly cited values for $T_2^a$ and $T_2^b$ are 20 and 80 ms, respectively (27), which leads to TE_opt ≈ 13.6 ms; however, this value drops closer to the typically used TE = 10 ms for smaller values of $T_2^a$ as seen in experimental studies (22). Also, although the TE_opt increases with increasing $T_2^a$ and decreasing $T_2^b$, the $\mathit{SNR}(M_o^a)\text{v}$. TE function also becomes more broad, so there is less at stake in optimizing the TE.

It is also important to note that the CRLB-derived values of TE_opt are not necessarily practical for any given imaging application. At low BW, imaging artifacts due to background field variation and chemical shift may limit the ability to effectively utilize the TE_opt. Also, depending on the application and hardware limitations, different T_const values may need to be considered. Longer T_const values will necessarily dictate longer TE_opt and CRLB calculations should be repeated for a condition where T_const is much different that 5 ms, as used herein. Lastly, the model assumed real data with additive Gaussian noise; however, at low SNR, the noise in magnitude MRI is Rician. For most cases of multi-exponential characterization, high SNR is required so the effect of noise fold-over in magnitude images is minimal. In general, though, one can correct for the effects of Rician noise on the echo magnitudes prior to data analysis (28), which will make the data analysis consistent with the CRLB calculations herein.

APPENDIX

Starting with Eq [2], and ignoring the noise term, the partial derivatives are

$$\frac{\partial M_T}{\partial M_0^a}=c_1^n\frac{\partial M_T}{\partial M_0^b}=c_2^n\frac{\partial M_T}{\partial T_2^a}=n{c}_3{c}_1^n\frac{\partial M_T}{\partial T_2^b}=n{c}_4{c}_2^n$$ [A.1]

where

$$c_1=e^{-TE/T_2^a}{c}_2=e^{-TE/T_2^b}{c}_3=\frac{TE{M}_0^a}{{(T_2^a)}^2}{c}_4=\frac{TE{M}_0^b}{{(T_2^b)}^2}.$$

Substituting this into Eq [4], taking the number of echoes to infinity, and using the series formulae

$$\begin{array}{l}\sum _{n=1}^{\infty }{r}^n=\frac{r}{1-r},\sum _{n=1}^{\infty }n{r}^n=\frac{r}{{(1-r)}^2},\text{and}\sum _{n=1}^{\infty }{n}^2{r}^n=\frac{r^2+r}{{(1-r)}^3},\text{we}\text{get}\\ F=\frac{1}{{\sigma }^2}\left[\begin{array}{cccc}\frac{c_1^2}{1-c_1^2}& & & \\ \frac{c_1{c}_2}{1-c_1{c}_2}& \frac{c_2^2}{1-c_2^2}& \text{N}& \\ \frac{c_1^2{c}_3}{{(1-c_1^2)}^2}& \frac{c_1{c}_2{c}_3}{{(1-c_1{c}_2)}^2}& \frac{(c_1^4+c_1^2)c_3^2}{{(1-c_1^2)}^3}& \\ \frac{c_1{c}_2{c}_4}{{(1-c_1{c}_2)}^2}& \frac{c_2^2{c}_4}{{(1-c_2^2)}^2}& \frac{(1+c_1{c}_2)c_1{c}_2{c}_3{c}_4}{{(1-c_1{c}_2)}^3}& \frac{(c_2^4+c_2^2)c_4^2}{{(1-c_2^2)}^3}\end{array}\right].\end{array}$$ [A.2]

Eq [3] then gives

$$\begin{array}{l}{s}^2(M_0^a)=\frac{-{\sigma }^2}{c_1^4{(c_1-c_2)}^6}(c_1^2-1){(c_1{c}_2-1)}^2\left(\begin{array}{l}{c}_2^2+2c_1{c}_2(c_2^2-3)+c_1^2(9-3c_2^2+c_2^4)-4c_1^3(c_2+c_2^3)\\ +c_1^4(-11+21c_2^2-3c_2^4)+c_1^5(2c_2-6c_2^3)+c_1^6(4-7c_2^2+4c_2^4)\end{array}\right)\\ s^2(M_0^b)=\frac{-{\sigma }^2}{c_1^4{(c_1-c_2)}^6}{(c_1{c}_2-1)}^2(c_2^2-1)\left(\begin{array}{l}{c}_1^2+2c_1(-3+c_1^2)c_2+(9-3c_1^2+c_1^4)c_2^2-4c_1(1+c_1^2)c_2^3\\ -(11-21c_1^2+3c_1^4)c_2^4+(2c_1-6c_1^3)c_2^5+(4-7c_1^2+4c_1^4)c_2^6\end{array}\right)\\ s^2(T_2^a)=\frac{-{\sigma }^2{(c_1^2-1)}^3{(c_1{c}_2-1)}^4}{c_1^4{(c_1-c_2)}^4{c}_3^2}\\ s^2(T_2^b)=\frac{-{\sigma }^2{(c_1{c}_2-1)}^4{(c_2^2-1)}^3}{{(c_1-c_2)}^4{c}_2^4{c}_4^2}\end{array}$$ [A.3]