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In medical applications of MRI the signal measured comes from multiple chemical species. For the purposes of this paper we will assume that the tissue is primarily composed of two chemical species, water and fat. The major motivation of this work is that the fat signal can obscure underlying pathology in the water image, which generally contains the majority of the diagnostic information. The most common approach to suppress the fat signal is fat saturation. This technique works well in regions where the magnetic field (B0) is homogeneous but can fail where the field is inhomogeneous (1). Dixon imaging, a technique that uses images collected at different times measured from the echo (echo time shifts) to separate water from the fat, can be significantly less sensitive to magnetic field (B0) inhomogeneities (2, 3). We are interested in optimizing the noise behavior for estimating both fat and water.
The original method proposed by Dixon used two images (points) and assumed a homogeneous magnetic field (2). This approach was generalized by Glover to handle field inhomogeneities by using three points (3, 4). The choice of echo time shifts for both of these approaches was limited by the availability of analytic techniques to solve for the fat and water images. The factors that may affect the variance of the estimates of water and fat include the echo time shifts (4, 5), field map (4, 6), and reconstruction algorithm (5). An algorithm was recently developed for data with arbitrary echo time shifts (7, 8). Here we provide a way of choosing the echo time shifts that produce the estimates of water and fat with minimum variance. In order to show the dependence of the optimization on the imaging sequence, we study the noise performance using spin echo (SE) or fast spin echo (FSE), spoiled gradient echo (SPGR), and steady-state free precession (SSFP).
The Cramér–Rao bound (CRB) (9, 10) is the lower bound on the variance of any unbiased estimate. If an algorithm produces the correct estimate in the absence of noise, then in the presence of noise, the variance of that estimate cannot be lower than the CRB. The CRB provides a measure of the minimum uncertainty of the estimates for a given data acquisition, independent of the reconstruction algorithm. The CRB can be used to optimize the imaging parameters, such as the choice of echo time shifts, and to compare the relative performance of different reconstruction algorithms with respect to the theoretical minimum variance.
In this paper we optimize the choice of echo time shifts to minimize the variance of the estimates of the water and fat magnitude and phase, as well as the field map. Previous noise analysis (4, 7), which did not account for the uncertainty in the field map propagating to the magnitude of the water and fat images, results in a prediction that the noise in these images is independent of fat:water ratio. The formulation of the CRB including the field map leads to a simple way to see that the noise performance is dependent on the fat–water composition. The nonlinear nature of the estimation leads to counterintuitive but experimentally verifiable results.
Obtaining fat–water separation with uniform noise properties is important because changes in the noise can mimic pathology, degrade image quality, and obscure relevant clinical information. Noise with unexpected properties has been observed clinically at interfaces between fat and water, in bone marrow, and in subcutaneous fat (8). In order to explain and avoid these artifacts it is necessary to include the field map in the noise analysis.
The Signal Model
The approach to chemical species separation is based on modeling the tissue as multiple species with a known spectra at discrete resonant frequencies. By acquiring images at multiple echo time shifts, one can estimate the individual species. The general approach can be used for more than two species (8, 11), extended to consider T2* decay (4) and peaks with variable width (4). Furthermore, the approach can take into account prior information or assumptions to reduce the number of images used (2, 12). While the CRB can be used to study all these generalizations, we have chosen to consider the typical case of two species, water and fat with single line spectra, in the presence of inhomogeneities in the magnetic field.
For the general problem where one needs to estimate the magnitude and phase of fat and water as well as the field map, there are five scalar unknowns. Therefore, we need at least five independent measurements, which leads us to the three-point water–fat decomposition method, which collects six scalar measurements.
The model for the signal within a voxel in terms of magnitude and phase is given by
where s(k) is the kth complex measurement. ρw and ϕw are the magnitude and phase of the water vector at the echo, ρf and ϕf are the magnitude and phase of the fat vector at the echo, Δw is the chemical shift between water and fat, tk is the time of the kth image measured from to the echo (echo time shift), Δϕk = Δwtk is the additional phase shift between fat and water caused by the difference in resonance frequency, Δψk = ψtk is the phase shift due to the field inhomogeneity (ψ), and ε is the noise (Gaussian distributed, zero mean and uncorrelated (13)).
In our model, we will consider the various imaging techniques by varying the alignment of the water and fat signals at the echo and restricting the realizable echo time shifts. In the case of SE or FSE, also known as TSE, the echo (t = 0) occurs at the spin echoes and the fat and water are aligned (14–16) (Fig. 1). The echo time shifts for SE or FSE can be positive or negative. In SPGR, also known as T1 FFE or FLASH, the echo (t = 0) occurs at the time of excitation and the fat and water are aligned. For SPGR, only positive echo time shifts are possible. In balanced SSFP, also known as balanced FFE or true FISP, the echo (t = 0) occurs in the midpoint of RF pulses (TR/2) and the water and fat spins are either aligned or anti-aligned, with most of them being anti-aligned (17, 18). Echo time shifts for SSFP can be either positive or negative.
The Cramér–Rao Bound
At the heart of the CRB is the Fisher information matrix (FIM) (9, 10). It can be interpreted as the sensitivity of the data to the parameters being estimated taking into account the noise,
where s is the vector containing the data, p is the vector containing the parameters of the model and Pr(s|p) is the probability of observing s given p.
The CRB is a bound on the covariance matrix of any unbiased estimator. The CRB is the inverse of the FIM,
where p̂ is any unbiased estimator of the parameters. This inequality should be interpreted in a matrix sense, implying that the difference (Cp̂ − F−1) is positive semidefinite (20). In particular, the CRB implies that the variance of any unbiased estimator (p̂k) cannot be smaller than the corresponding diagonal component of the inverse of the FIM:
The CRB has been applied to a wide range of problems in medical imaging. In MRI it has been used in the estimation of the magnitude, phase, and variance of the image (21) and diffusion coefficient (22). The CRB has a natural connection to maximum likelihood estimation (MLE) since if an unbiased estimator exists that achieves the CRB, it will maximize the likelihood. Because least-squares estimation is the MLE for Gaussian noise, the CRB for estimating complex fat and water has already been presented in Ref. (7) for a homogeneous magnetic field. A preliminary result regarding the estimation of the real and imaginary components of fat and water for an unknown field map was previously presented (23). The theoretical contribution in this paper is the inclusion of an unknown field map, which allows us to model the effect of the fat:water ratio in the variance of magnitude, phase, and field map. We then use the CRB to optimize the echo time shifts.
The noise efficiency of fat and water estimates has traditionally been quantified by the effective number of signals averaged (NSA) defined as (4)
where is an estimate of the magnitude and σ is the variance in each of the measured images.
This expression of the noise efficiency for the magnitude varies from 0 to the number of images (N), in our case 3. If NSA = 0, it implies that fat or water cannot be estimated in a stable way from the data. A NSA = 3 implies a noise efficiency equivalent to estimating the mean from three independent noisy samples, which is the maximum possible. If an unbiased estimator achieves a NSA = 3, then it is efficient. This normalization by the variance of the measurements provides a way to compare across techniques with different number of images and is independent of their variance.
This formalism works so long as the only quantities that are estimated are the magnitudes of fat and water, which are generally the parameters of interest. This normalization does not extend to the estimates of the phase or field map. To express the noise properties of all parameters in a consistent way, we generalize by dividing the minimum variance of the parameter given that it is the only unknown in the model by the variance of the estimate:
We have five unknowns our model (Eq. ), which for notational simplicity we put in a vector p with elements p1 = ρw, p2 = ρf, p3 = ϕw, p4 = ϕf, and p5 = ψ. This generalization of the NSA matches the previous formulation but also gives meaningful numbers for parameters that are not linearly related to the data. The following bounds give the minimum variance for estimating each of the parameters with N images where all other parameters in the model are known:
The definition of the NSA given by Eq.  ranges from 0 and N for all parameters including the phase and the field map. These CRB normalizations for magnitude and phase have already appeared in Ref. (21) but to our knowledge this is the first published expression for the CRB for estimating the field map. To obtain the maximum NSA when all of the parameters are unknown, we substitute σwith [F−1]kk in Eq. .
METHODS AND RESULTS
Fisher Information Matrix
The Fisher information matrix for the three-point decomposition of fat and water is a 5×5 matrix, which can be inverted numerically to obtain the CRB. We write the data from three echo time shifts as a matrix equation using real parameters and measurements. Rewriting the signal equation in this way simplifies the computation of the FIM since otherwise two of the parameters are complex (the fat and water densities) and the third is real (the field map). Writing the signal model for three measurements in matrix form we get
The subscripts refer to the measurement index and the superscripts to whether the quantity represents the real or imaginary component of the signal.
To derive the FIM we use Eq.  and compute the elementwise derivatives. Then we rewrite it in terms of the system matrix A:
We use the symmetry of the FIM to fill in the missing terms, F(k = 3, 4, 5, l = 1, 2). Details of the derivation of this expression are given in the Appendix.
The nonlinearity of the estimation problem leads to an expression for the FIM that depends on the magnitude of water and fat. In the field-known case, the FIM is independent of chemical composition of the pixel.
Optimization of the Echo Time Shifts
There are many figures of merit that could be used to choose the echo time shifts. We have chosen to maximize the minimum attainable NSA across all ratios of fat:water densities. This approach weighs the worst case scenario heavily since spurious noisy pixels can degrade the diagnostic quality of an image if they are at critical locations. Among the echo time shifts that achieve a given NSA, those which require less imaging time are preferable. Since the imaging time is related to the largest echo time shift, we focus our optimization on finding the echo time shifts that maximize the NSA with the minimum phase difference between fat and water (Δϕk) as a surrogate for time.
The theoretical result in Eqs. [13–15] allows us to compute the maximum NSA for a given choice of echo time shifts and a fixed chemical composition of the pixel. To carry out the optimization, we first find the minimum of the attainable NSA for fat:water ratios ranging from 0.01 to 100 (keeping the sum of the water and fat densities equal to 1) for all possible echo time shifts less than 4π (with echo time shift increments of 2π/100). To carry out this computation of the inverse of the FIM in a numerically stable way for all echo time shifts, we set the NSA to 0 for all FIM with a condition number larger than 1012. We also added a perturbation of 10−6 to the water density to avoid the numerical instability when W = F. We then find the maximum for acquisitions where all the echo time shifts equal or less than Δϕk to produce a plot of MAX (MIN NSA) vs. Δϕk. By construction this plot is monotonically increasing and the first Δϕk for which it stops increasing will be the shortest time that gives the best NSA.
This optimization for the magnitudes of fat and water leads to Fig. 2. This figure contains a new result with respect of the ideal choice of echo time shifts. Previous NSA analysis for the field-known (linear) case (4, 7, 15) leads to an optimum choice of echo time shifts of (−2π/3, 0, 2π/3). This solution is ideal if the pixel only contains water or fat. The choice of (−π/6, π/2, 7π/6) only arises when we consider the nonlinear problem. This choice of echo time shifts is ideal for any fat:water ratio. In the case of SPGR, we shift the entire acquisition by π to achieve the ideal noise performance. The ideal acquisition for the estimation of the magnitude can be achieved with echo time shifts of the form (−π/6 + πk, π/2 + πk, 7π/6 + πk). Like the optimal acquisition for the field-known case, the measurements are equally spaced over the unit circle but instead of being symmetric about t = 0 (Fig. 1), they are shifted by π/2 + πk. The theoretical results presented here have been verified with phantom experiments and clinically (8).
The results for the optimization of the NSA for phase estimation are presented in Fig. 3. To the best of our knowledge this paper presents the first analysis of noise in the phase of imaging methods that decompose water and fat. A current application of phase estimation is thermometry (19). It is interesting that optimizing the magnitude images does not optimize the phase images. Using the magnitude images to choose the echo time shifts, as the field-known result would suggest, leads to suboptimal phase estimates. The echo time shifts that require the least time to achieve the maximum NSA for the phase estimate are (−2π/3, 0, 2π/3). The noise properties of the phase are not invariant to shifts of 2π. In particular, the SPGR sequence at (4π/3, 2π, 8π/3) does not lead to the ideal noise performance.
Figure 4 shows the results for field map estimation. Most often the field map is a nuisance parameter that must be estimated but does not contain diagnostic information. There are some applications when it becomes useful as a way of reducing the number of points needed for subsequent fat–water decompositions (12). The echo time shifts that require the least time to achieve the maximum NSA are (−π, 0, π). We again see that as in the case of phase estimation, SPGR is unable to achieve a NSA of 3 (i.e., efficiency).
Monte Carlo Experiments of NSA Dependence on Fat:Water Ratio
The results presented so far are theoretical bounds on the noise performance that may not be achieved by a practical estimator (i.e., a reconstruction algorithm). This section tests a version of the nonlinear least-squares estimator used in Refs. (7, 8) and in the process verifies that the CRB is attainable. The difference from what is used in Refs. (7, 8) lies in the smoothing of the field map. In the following Monte Carlo simulations, no field map smoothing was used. Since field map smoothing effectively introduces prior information about the field map into the estimation, it also introduces a bias in the cases where the field map does not have the smoothness assumed by the algorithm. While the effect of prior information on the bias-variance trade-off is important, it is outside the scope of this paper.
Monte Carlo simulations were generated with ϕw = ϕf = π/4, ψ = π/20, SNR = 200, and 500 independent Gaussian noise realizations for every fat: water ratio. The fat:water ratio was varied from 0.01 to 100 such that the samples were uniformly spaced on a logarithmic scale. In the case of SSFP, a π phase shift was added to the fat signal at t = 0.
For the ideal choice of echo time shifts for magnitude estimation (−π/6, π/2, 7π/6), shown in Fig. 5, the Monte Carlo estimates of all quantities matched the theoretical bound, implying that the estimator is efficient for these parameters. Even though this choice of echo time shifts has optimal (and constant) NSA for the magnitude at all fat:water ratios, it has suboptimal and structured behavior for the phase and field map. Because of its optimality in magnitude estimation, this choice of echo time shifts has been chosen for use in iterative decomposition of water and fat with echo asymmetry and least squares estimation (IDEAL) imaging (8). The noise behavior for magnitude estimates is identical for (−π/6 + πk, π/2 + πk, 7π/6 + πk), with integer k, but the phase and field map behavior is different (Fig. 6).
The ideal choice of echo time shifts for magnitude estimation in the field-known case (−2π/3, 0, 2π/3), is the ideal choice for phase estimation (Fig. 7). In terms of the magnitude, the simulation verifies the lack of estimability of fat and water when the amounts of fat and water are equal (5, 8). This inability to estimate fat and water at equal amounts leads to clinically observable “salt and pepper” noise at water–fat interfaces (8). The asymmetry of the variance of the magnitude depending on whether the the tissue is mostly water or fat has also been observed clinically. The CRB predicts that the water magnitude image will be twice as noisy in the regions that are mostly fat. This is consistent with clinical observations of images of the bone marrow and subcutaneous fat (8).
With respect to the phase, the least-squares estimation matches the CRB everywhere except where the magnitude of the species being estimated is small or when the magnitude is not estimable. This behavior is not fundamental but brought about by the particular estimation algorithm, the SNR, and the amount of field inhomogeneity. The problem becomes more ill posed in these cases and the algorithm as implemented does not perform ideally under those conditions. As we lower the noise in our simulations (not shown), the Monte Carlo results matched the CRB perfectly even with this algorithm. The parameters for the Monte Carlo simulations were chosen because they illustrate the behavior of the estimation algorithm and because they highlight the advantage of having a CRB that is algorithm independent.
The NSA of the field map explains why the noise behavior is different for the field-known (4, 7, 15) and the current field-unknown analysis. When fat and water are in equal proportions with symmetric echo time shifts (−t, 0, t), the field map cannot be estimated. This uncertainty is propagated into the magnitude. A geometric explanation of this phenomenon has been presented elsewhere (5).
Figure 8 shows the NSA for the symmetric equally spaced acquisition, but this time for a case in which fat and water are π out of phase at t = 0 (modeling SSFP sequences). This change removes the ambiguity at equal amounts of fat and water both in magnitude and in field-map estimation. This difference in the behavior explains why, even though unappreciated at the time, the images in Ref. (6) did not show jagged edges at boundaries between fat and water. For small amounts of the estimated quantity, the estimator has a larger variance than the CRB. The larger variance for quantities in smaller proportions is understandable since the signal is dominated by the other species.
Field-map estimation with SE, FSE, or SSFP (Fig. 9) is optimal at echo time shifts (−π, 0, π), which has been previously used for magnitude estimation (3) but whose optimality with respect to field-map estimation has not been appreciated. For this choice of echo time shifts, all of the parameters achieve a constant NSA. The overall performance is very desirable. However, the CRB analysis presented here does not evaluate the robustness to small changes in the echo time shifts. Small perturbations around (−π, 0, π) lead to the singularity when fat and water magnitudes are equal (8). This was observed in the simulations, as seen in the plots for magnitude and phase exactly for the point where water and fat are equal (Fig. 9).
Field-map estimation using SPGR (Fig. 10) is optimal with a sequence that effectively takes two points at t = 0 and one at 2π, i.e., (0, 0.04π, 2.02π). The slight shift from t = 0 on the second measurement was only to avoid the FIM from becoming singular. This choice is optimal for estimation of the field map; however, the other two parameters cannot be estimated. Note that the normalization for the field map (Eq. ) includes the imaging time; therefore, while having the third echo time shift at 2π leads to the best normalized NSA, there may be longer times where the variance of the field map would be lower. It just would not be as efficient with respect to the minimum variance for that imaging time.
The primary result for magnitude estimation is the recommendation of equally spaced echo time shifts centered about a π/2 phase shift between fat and water (−π/6 + πk, π/2 + πk, 7π/6 + πk). In hindsight, this is an intuitive choice since the middle echo time shift in that acquisition would put water in the real channel and fat in the imaginary channel if there were no phase errors in the measurements. Its discovery, however, required including the field map in the noise analysis. Figure 11 shows the effect of the new choice of echo time shifts in clinical images of a knee. The images show a jagged edge for the symmetric acquisition at the boundary of muscle and fat but clearly defined boundaries for the asymmetric acquisition. A thorough presentation of its implications for imaging with FSE and its comparison to fat saturation is presented in the accompanying paper (8).
We have seen that SSFP, where fat and water have a π phase shift at t = 0, leads to better noise performance for symmetric acquisitions. Our analysis implies that a steady-state sequence with a π/2 phase shift between water and fat at the echo would be ideal when combined with a symmetric acquisition. Development of such an imaging sequence would help to reduce imaging time.
An interesting result is that the noise properties of the magnitude appear to be invariant to shifts of 2π in the echo time shifts while those of the phase are not. This can most strikingly be seen in the difference in NSA between FSE and SPGR sequences for phase estimation. While FSE is efficient for (−2π/3, 0, 2π/3), this choice is not allowed for SPGR. Adding a 2π shift, which leads to an acquisition allowed by SPGR, leads to suboptimal performance. It appears that acquisition of images with both positive and negative echo time shifts is necessary to eliminate the effect of the field map. This property was exploited by Glover and Schneider (3) to compute his analytic solution. This explanation, while consistent with observed results, is incomplete in creating a satisfactory intuition. The lack of an invariance to 2π shifts highlights the usefulness of CRB analysis over geometric intuition.
In the majority of applications, the field map is a parameter that must be estimated but has no clinical information, i.e., a nuisance parameter. Because of this, its noise properties are mostly ignored. However, it is important to consider it in order to understand the noise behavior of those parameters in which we are interested, for example, the singularity at equal amounts of fat and water for symmetric acquisitions. The field map is also important in imaging methods that try to reduce the number of points by using three-point water–fat separation to estimate the initial phase and field map, followed by repeated one-point acquisitions assuming those parameters are known (12). Proper estimation of the field map is also important in techniques that use field-map smoothing (25), since the optimal amount of smoothing will depend on the variance of the estimate of the field map.
Issues such as field-map smoothing and regularization of the inversion process lead to the trade-off between bias and variance. It is possible for a biased estimator to have less overall error than an unbiased one if its bias and variance are small. As the problem becomes more ill conditioned, as is the case for shorter imaging times (reduced echo time shifts), this ability to trade a lower variance for some bias becomes more important. In the same way that the estimation algorithm can be modified, so can the CRB. An area for future work is extending the CRB analysis to account for bias and to use it for algorithm optimization.
An important assumption in the use of the CRB as presented here is that the estimator is unbiased. As a representative justification of this assumption, we compared the sample mean of the estimates from the Monte Carlo simulation to the true value for equal amount of water and fat. For nearly all echo time shifts, to three significant figures, the mean from the simulation matched the true value. The only cases for which the mean estimate did not match the true value was where fat and water were in equal amounts for a symmetric echo time shifts. In that case, the estimate oscillated between the true value and zero. Another situation where the algorithm is biased is at low SNR for cases where the other species is dominant.
The main purpose of the Monte Carlo simulations presented in this paper is to show that the CRB can be achieved and to illustrate some of the differences from results that could be obtained from Monte Carlo simulations. The SNR = 200 was chosen for this purpose. This SNR avoids one of the major hurdles in nonlinear least-squares estimation: local minima in the functional being minimized. In our current clinical application of the algorithm, we use field-map smoothing. Although comprehensive discussion of field-map smoothing is beyond the scope of this paper, the behavior of the unsmoothed algorithm at an SNR = 25 shows interesting behavior and highlights the effect of local minima.
In Fig. 12, the NSA of both magnitude and phase is well above the upper limit set by the CRB when there is a small percentage of that species. These differences are beyond what can be justified by sampling errors and noise. Since the theory bounds any unbiased estimator, we suspected that the estimates must be biased. From the 500 independent noise realizations when fat:water ratio is 1/100, we found that the mean of the estimates for the magnitude and phase of fat were statistically different than the true value (P value of 0.001%). On the other hand, the magnitude and phase of water, as well as the field map, cannot be considered different than their true value (P value 10%). The practical implication of this bias is small in most applications since the error occurs in the signal that is in a small proportion in the pixel.
This paper focused on finding the choices of echo time shifts for individual parameters that achieve the maximum NSA in the least amount of time, but there are other criteria one could use to choose the echo time shifts. Using the expression for the CRB provided, one could optimize only the pixels with all water, for the average across all fat:water ratios or over a different range of practical echo time shifts. In particular, for optimization, it might be more intuitive to use the CRB itself or the SNR instead of the NSA. The normalization used to compute the NSA generates a convenient range of values to present the results and a simple way to compare technique choices but it does not compute the variance of the estimate in absolute terms. This would be relevant if the goal was to achieve a specific SNR. The normalization chosen can be particularly misleading for the field map since the normalization is a function of the echo time shifts. For example, Fig. 4 might be interpreted as saying that there can be no improvement in the noise performance for SPGR for echo time shifts greater than 2π. That would be true in terms of noise efficiency, but since the normalization for the phase includes time, it is possible that echo time shifts with longer imaging times could reduce the variance of the field-map estimate, although they would not be as noise efficient. There are also other considerations with respect to the choice of echo time shifts than those discussed in this paper. Increasing the echo time shifts may lead to unfavorable banding artifacts in SSFP or possibly to unacceptable blurring because of T2 decay. Under such circumstances our analysis becomes just one of the factors to consider. Some of those issues are discussed in Refs. (6–8).
The major advantage of using the CRB to study the noise behavior of an estimation problem is that its results are truly representative of the information in the data. Other choices, like Monte Carlo studies or experiments, are always subject to the limitations of the estimation algorithm. We hope this paper encourages others to use this technique for parameter and algorithm optimization. In its application to the decomposition of water and fat, the computation of the CRB reduces to the inversion of a 5×5 matrix, which is computationally fast. This allows for easy exploration of the parameter space. This analysis was important in choosing the echo time shifts for IDEAL imaging (7, 8) and showing that the nonlinear algorithm used was efficient in Monte Carlo and phantom experiments.
Our definition for the NSA used a normalization based on knowing all of the parameters except for the one being estimated. It is therefore a somewhat surprising result that there exists a choice of echo time shifts for which the estimation is efficient since it implies that the uncertainty of not knowing all of the other parameters does not propagate into the one being estimated.
Inclusion of the uncertainty in the field map into the noise analysis leads to a nonlinear estimation problem. This nonlinearity makes the noise properties of the estimates dependent on the values of the parameters themselves, for example fat:water ratio. The first observation of this dependence came from an early formulation of the FIM for estimation of the real and imaginary components (23), which helped illuminate the ambiguity when water equals fat. Intuition based on the geometric interpretation of the signal and previously used error propagation methods work well for linear estimation problems. One can expect that the CRB will have a significant impact when nonlinear estimation can be reduced to a few parameters.
We have generalized the noise analysis for for the decomposition of water and fat to include the uncertainty in the field-map estimation. This leads to a nonlinear estimation problem that results in new choices of echo time shifts for optimal noise performance including the dependence of the noise of the fat:water ratio. The mathematical tool used to optimize the choice of echo time shifts was the Cramér–Rao bound. Through Monte Carlo studies, we showed that the CRB is achievable using an iterative least-squares algorithm and that it appropriately describes the fundamental bound on the variance of the magnitude, phase, and field-map estimates.
Derivation of the Fisher Information Matrix
To derive the FIM (Eqs. [13–15]) from the definition (Eq. ) we begin with the probability density function of the measurements,
We then rewrite log of the probability in terms of our expression for the mean signal from Eq. ,
We begin by deriving the part of the FIM relating solely to the magnitudes of fat and water (F(k = 1, 2, l = 1, 2)). The partial derivatives are computed by expanding the matrix multiplications into sums, evaluating the derivatives, and then rewriting the results as matrix multiplications:
Hence, F(k = 1, 2, l = 1, 2) = [AtA]. Note the negative sign in the definition of the FIM and that this result reproduces the expression used when the field map is known (7).
The following section refers to the effect of the phase and the field map on the uncertainty of the magnitude estimates (F(k = 1, 2, l = 3, 4, 5)). Note that because of our ordering of the parameters, A is a function of pl but not pk. By deriving Eq.  we obtain
At this point, we need to compute the expectation over the noise realizations, which allows us to substitute Aρ for s̄:
Therefore, Finally, the last part of the the FIM, which relates to how the phase and field map affect each other (F(k = 3, 4, 5, l = 3, 4, 5)), is derived in the same manner but noting that now the matrix A is a function of both pl and pk:
Hence, F(k = 3,4,5,l = 3,4,5)= . For the simplification we used the facts that if the second derivative exists, then and that for any N×1 real vectors x and y, xty= ytx. For example, .