Effects of Image Noise in Muscle DT-MRI Assessed Using Numerical Simulations

Abstract

Diffusion-tensor MRI (DT-MRI) studies of skeletal muscle provide information about muscle architecture, microstructure, and damage. However, the effects of noise, the diffusion weighting (b-) value, and partial volume artifacts on the estimation of the diffusion tensor (D) are unknown. This study investigated these issues using Monte Carlo simulations of 3×9 voxel regions of interest containing muscle, adipose tissue, and intermediate degrees of muscle volume fractions (f_M). 1000 simulations were performed for each of 8 b-values and 11 SNR levels. The dependences of the eigenvalues (λ_1-3), mean diffusivity ( $\stackrel{‒}{\lambda }$ ), fractional anisotropy (FA), and the angular deviation of the first eigenvector from its true value (α) were observed. For moderate b-values (b=435-725 s/mm²) and f_M=1, an accuracy of 5% was obtained for λ_1-3, $\stackrel{‒}{\lambda }$ , and FA with an SNR of 25. An accuracy of 1% was obtained for λ_1-3, $\stackrel{‒}{\lambda }$ , and FA with f_M=1 and SNR=50. For regions with f_M=8/9, 5% accuracy was obtained with SNR=40. For α, SNR≥25 and ≥45 were required for ±4.5° uncertainty with f_M=1 and f_M=0.5, respectively; SNR≥60 was required for ±9° uncertainty in single muscle voxels. These findings may influence the design and interpretation of DT-MRI studies of muscle microstructure, damage, and architecture.

INTRODUCTION

Diffusion-tensor MRI (DT-MRI) has emerged as a promising tool for in vivo studies of muscle architecture (1-5), damage and inflammatory processes (6-10), microstructure (11-15), and for quantitative interpretation of blood-oxygenation level-dependent (BOLD) contrast effects (16-18). Several recent quantitative muscle architecture studies have used DT-MRI based muscle fiber tractography (1,2,4). The capability for DT-MRI muscle fiber tracking is based on the preferential diffusion of water along the longitudinal axis of the muscle fibers, as first identified by Cleveland et al. (19) and subsequently confirmed in studies of both skeletal (20,21) and cardiac (22,23) muscle. This technique may potentially be used to provide essential input data for biomechanical models that elucidate muscle structure-function relationships.

Several studies have demonstrated that measuring diffusion transverse to the muscle fibers’ long axes may be useful for examining the health and microstructural properties of muscle. For example, Heemskerk et al. have shown that elements of the muscle diffusion tensor (D), and in particular its third eigenvalue (λ₃), are correlated with histological damage indices following ischemia-reperfusion injury (6); Saotome et al reported different temporal responses of the second eigenvalue (λ₂) and λ₃ to nerve transection (12). Moreover, DT-MRI appears to reflect damage and recovery processes distinctly from T₂ (7). In human DT-MRI studies, muscle damage is associated with lower fractional anisotropy (FA; (8)). In addition, the transverse diffusivities appear to be sensitive to gender (13) and age (14). A common explanation for these findings may lie in microstructural and microfunctional alterations such as muscle fiber atrophy or increased membrane permeability; in addition, muscle damage-induced diffusivity changes may reflect the addition of distinct T₂ and diffusion components related to inflammation (9).

Muscle DT-MRI is made challenging by low T₂ (∼35 ms at 3T); long T₁ (∼1200 ms at 3T); high diffusivities that create large signal loss (first-third eigenvalues of λ₁≅2.1×10^-3 mm²/s, λ₂≅1.7×10^-3 mm²/s, and λ₃≅1.4×10^-3 mm²/s (3,8,13,24,25)); low FA (∼0.2 – ∼0.4) (3,8,13,24,25); and difficult-to-shim volumes. Clearly, achieving an adequate signal-to-noise ratio (SNR) is essential to estimating D in muscle accurately. A fundamental difficulty, however, is the lack of a complete understanding of what constitutes an adequate SNR for muscle DT-MRI.

The SNR dependence of DT-MRI in white matter structures has been examined using theoretical, simulation, and experimental approaches (26-31). With low levels of diffusion weighting (as reflected in the b-value), a pattern of λ₁ overestimation and λ₂ and λ₃ underestimation, termed eigenvalue repulsion, has been observed both experimentally (29) and in simulations (26-28,31). At high b-values, however, baseline offsets due to Rician-distributed noise can cause diffusion coefficients to be underestimated (30,31); a transition from over- to under- estimation of anisotropy occurs with increasing b-values as well (31). The details of the b-value dependence of tensor estimation notwithstanding, a general conclusion to be drawn from the above-cited studies is that an SNR>20 in the T₂-weighted image is adequate for accurate specification of λ_1-3 and an SNR>5 is adequate for specification of the trace of D. However, as compared to neuronal DT-MRI, the higher diffusivities in muscle may cause greater signal loss in the diffusion-weighted images and consequently more stringent SNR requirements. Therefore, the first purpose of this study was to test the dependence of estimating D in muscle and its derived indices on SNR and the b-value.

Another complication in muscle DT-MRI is that adipose tissue infiltrates may occur in obesity, healthy aging, in diseases such as the dystrophies and inflammatory myopathies, and following injury. These infiltrates may have a relatively homogeneous spatial distribution, such as following ischemic injury (32), or in an inhomogeneous spatial pattern, such as fat infiltration along interfascicular lines in obesity. In either case, the finite resolution of MRI results in adipose-muscle partial volume artifacts that may alter the apparent diffusion properties within specified regions of interest (ROIs). Contamination of D by signals from lipids may also result from chemical shift artifacts: while the lipid methyl resonance is suppressed in common DTI acquisition methods such as EPI, some residual chemical shift artifact may occur. In fiber tracking applications, these artifacts may result in a premature termination or inaccurate specification of fiber tracts; also, λ_1-3 will experience some degree of bias by fat signal contamination. Therefore, the second purpose of this study was to determine the dependence of the estimation of D on the degree of partial volume artifact, at different levels of b and SNR.

METHODS

General

Numerical simulations were conducted using Matlab v. 7.01.24704, R14 (The Mathworks, Inc., Natick MA). The simulations were conducted in three steps: definition of the model muscle; calculation of diffusion-weighted images at various levels of b; and estimation of D at various levels of SNR, including statistical analysis of the components of D and their derived indices. One thousand Monte Carlo simulations were performed for each of eight b-values at each of eleven SNR levels.

Definition of the Model Tissue

A model tissue containing 3×9 muscle voxels and 3×9 fat voxels (3×18 total size) was defined. These model tissue sizes differ slightly from those used in previous studies (e.g., 5×5 in Ref. (28)), but this design facilitated the analysis of partial volume artifacts described below. Table 1 shows the NMR and diffusion properties assigned to each tissue type. For each muscle voxel, the eigenvector corresponding to λ₁ (ε₁) was specified in the +X (horizontal) direction and ε₂ and ε₃ were specified in the +Y (vertical) and +Z (out-of-plane) directions, respectively. For individual adipose tissue elements, ε_1-3 were randomly assigned to lie along either the X, Y, or Z directions with equal probability (i.e., isotropic diffusion).

Table 1.

NMR and diffusion properties for the model muscle and fat tissues. Units are as follows: Proton density, arbitrary units; T₁ and T₂, ms; diffusivities, ×10^-3 mm²/s. FA is dimensionless.

	Tissue Type
Property	Muscle	Fat
Proton Density (ρ)1	0.80	0.10
T1	12002	3002
T2	35	80
λ 1	2.13	0.63
λ 2	1.7	0.6
λ 3	1.4	0.6
FA	0.20	0.0

¹Proton density values assume 80% muscle water content, 10% adipose water content, and the use of a fat suppression pulse.

²Muscle and fat relaxation time constants are based on measured values at 3T (data not shown).

³Muscle diffusivities are based on Refs. (3,8,13,24,25); fat diffusivities are based on Ref. (10).

Calculation of Diffusion Weighted Images

A T₂-weighted image was calculated according to:

$$S=\rho \cdot (1-e^{-TR∕T_1})\cdot e^{-TE∕T_2}$$ [1]

where S is the (non-diffusion-weighted) signal intensity, ρ is the proton density, and TR and TE are the repetition and echo times respectively. The values for TR and TE (5000 and 45 ms, respectively) were similar to those used in our previous muscle DTI studies (1,4). The signal intensities were divided by the muscle signal intensity, so that muscle voxels had S=1 and fat voxels had S=0.2617. For each model element, D was calculated as:

$$\mathbf{D}=\mathbf{E}\Lambda {\mathbf{E}}^T$$ [2]

where the superscript T indicates the transpose operation, E is a 3×3 matrix composed of ε_1-3:

$$\mathbf{E}=\mid \begin{array}{ccc}\hfill {\epsilon }_{1X}\hfill & \hfill {\epsilon }_{2X}\hfill & \hfill {\epsilon }_{3X}\hfill \\ \hfill {\epsilon }_{1Y}\hfill & \hfill {\epsilon }_{2Y}\hfill & \hfill {\epsilon }_{3Y}\hfill \\ \hfill {\epsilon }_{1Z}\hfill & \hfill {\epsilon }_{2Z}\hfill & \hfill {\epsilon }_{3Z}\hfill \end{array}\mid ,$$ [3]

and Λ is a 3×3 matrix composed of λ_1-3 along the main diagonal:

$$\Lambda =\mid \begin{array}{ccc}\hfill {\lambda }_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\lambda }_2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\lambda }_3\hfill \end{array}\mid .$$ [4]

A diffusion weighting matrix R was defined as

$$\mathbf{R}=\mid \begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \sqrt{1∕2}\hfill & \hfill \sqrt{1∕2}\hfill & \hfill 0\hfill \\ \hfill \sqrt{1∕2}\hfill & \hfill 0\hfill & \hfill \sqrt{1∕2}\hfill \\ \hfill 0\hfill & \hfill \sqrt{1∕2}\hfill & \hfill \sqrt{1∕2}\hfill \end{array}\mid .$$ [5]

The diffusion-weighted signal, S_D, was calculated for each of the six directions specified in R as:

$$S_D=S\cdot e^{-b\cdot \mathbf{r}\cdot \mathbf{D}\cdot {\mathbf{r}}^T},$$ [6]

where the subscript D reflects the direction specified by r, a 1×3 vector corresponding to each row of R. The images were calculated for b-values of 145, 290, 435, 580, 725, 870, 1015, and 1160 s/mm². The b-value of 580 s/mm² was chosen because it is approximately equal to the theoretical optimum ( $=\frac{1.11}{\stackrel{‒}{\lambda }}$ ) (33), where $\stackrel{‒}{\lambda }$ is the one-third of Trace(D). A large range of b-values was used in order to account for the reduction in the optimum value due to T₂ effects (34), to test the expected dependence in the parameters’ SNR sensitivities on b (31), and to encompass the full range of b-values that have been reported in the muscle DT-MRI literature (400–927 s/mm²). The highest b-value of 1160 s/mm² was approximately equal to the T₂-adjusted optimum for adipose tissue.

Calculation of D and Derived Indices at Various Levels of SNR

Noise was added to the images and D was estimated by using the following procedures. Zero mean, unit variance noise was created in quadrature, divided by scaling factors of 5, 10, 15, 20, 30, 40, 60, 80, 100, 120, and 140, and added to the images; then the magnitudes of the image signals were calculated. The mean values of the T₂-weighted and six diffusion-weighted signal intensities in the 27 muscle-only pixels were calculated and used to form a signal vector S. A weighted least-squares fit (35) was performed in order to estimate D. To do so, a 7×7 design matrix B was defined, with the first row equal to

$${\mathbf{B}}_1=\mid 1000000\mid$$ [7a]

and each additional row i given by

$${\mathbf{B}}_i=\mid 1-x_{(i-1)}^2-y_{(i-1)}^2-z_{(i-1)}^2-2x_{(i-1)}{y}_{(i-1)}-2x_{(i-1)}{z}_{(i-1)}-2y_{(i-1)}{z}_{(i-1)}\mid ,$$ [7b]

where x_(i-1), y_(i-1), and z_(i-1) refer to the X, Y, and Z diffusion weighting coefficients in row (i–1) of R. Based on the assumption of equal variance in all images (which follows directly from the method for defining the noise in the simulated images), a weighting matrix W was defined as a 7×7 matrix composed of S along the main diagonal. By defining the weighting matrix in this way, the estimation of D is weighted towards the data from images with higher SNR. A 1×7 vector of regression coefficients, A, (with an intercept ${\widehat{\beta }}_0$ and 6 slopes ${\widehat{\beta }}_{1-6}$ ) was calculated as:

$$\mathbf{A}={\left({\mathbf{M}}^T\mathbf{M}\right)}^{-1}{\mathbf{M}}^T\mathbf{WS}\prime ,$$ [8]

where M is the weighted design matrix ( M = WB ) and S’ is used to indicate the natural log transformation of S. D was then formed by:

$$\mathbf{D}=\frac{1}{b}\cdot \mid \begin{array}{ccc}\hfill \mathbf{A}\left(2\right)\hfill & \hfill \mathbf{A}\left(5\right)\hfill & \hfill \mathbf{A}\left(6\right)\hfill \\ \hfill \mathbf{A}\left(5\right)\hfill & \hfill \mathbf{A}\left(3\right)\hfill & \hfill \mathbf{A}\left(7\right)\hfill \\ \hfill \mathbf{A}\left(6\right)\hfill & \hfill \mathbf{A}\left(7\right)\hfill & \hfill \mathbf{A}\left(4\right)\hfill \end{array}\mid .$$ [9]

The eigenvalues and eigenvectors were obtained by diagonalization using the eigs function in Matlab, which performs a magnitude sorting of λ_1-3. $\stackrel{‒}{\lambda }$ and FA were calculated, the latter as:

$$FA=\frac{\sqrt{3}}{\sqrt{2}}\cdot \frac{\sqrt{{({\lambda }_1-\stackrel{‒}{\lambda })}^2+{({\lambda }_2-\stackrel{‒}{\lambda })}^2+{({\lambda }_3-\stackrel{‒}{\lambda })}^2}}{\sqrt{{\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2}}.$$ [10]

The SNR and b-value dependences of the diffusion ellipsoid orientation estimate were evaluated by measuring the fraction of times that ε₁ was correctly oriented nearest the X axis (f_E1X) and by measuring the angular deviation (α) of ε₁ from X, a unit column vector in the +X direction. If necessary (i.e., if the X component of ε₁ was negative), the sign of ε₁ was reversed. Then α was calculated as:

$$\alpha =\mathrm{acos}({\epsilon }_1\cdot \mathbf{X}).$$ [11]

In order to provide relevance to fiber tracking applications, the SNR and b-value dependences of f_E1X and α were also assessed in a single muscle voxel from each of the 88,000 simulations. To measure the effect of muscle volume fraction (f_M) on the apparent diffusion properties of ROIs, all measurements were also performed after shifting the ROI by progressively greater amounts into the adipose region of the model tissue. This resulted in f_M values ranging from 9/9 to 0/9, in steps of –1/9. Finally, to provide insight into the behavior of λ₁ in the low SNR regime (defined as muscle SNR<40 or adipose SNR<20), the diffusion-weighted signal in the X direction, S_X, was plotted as a function of b-value at various levels of SNR.

Statistical Analysis

The dependences of λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA on f_M, b, and SNR were analyzed in two ways: graphically, so that overall trends could be noted; and quantitatively, by determining the mean and standard deviation (SD) for the estimates at each f_M, b, and SNR value. Because α was expected to be distributed symmetrically about +X with both positive and negative values and a mean of zero, its uncertainty was calculated as the SD of its distribution. f_E1X is inherently a population parameter and so no statistics were calculated. The accuracy of the measurements was defined as the difference between the mean value for a given SNR and b-value and the programmed value. To quantify the SNR required for accuracy levels of 10, 5, and 1% in muscle tissue, the λ₁, λ₂, λ₃, and $\stackrel{‒}{\lambda }$ data were analyzed to determine the SNR corresponding to point at which the mean value first intersected the regions of ±10%, ±5%, and ±1% of the true value (SNR_10%, SNR_5%, and SNR_1%, respectively). For α, the SNR required for the SD to intersect the regions 0-9°, 0-4.5°, and 0-0.9° was calculated for both the ROI and single-voxel analyses. Generally, it was necessary to obtain SNR_10%, SNR_5%, and SNR_1% by linear interpolation between adjacent data points. Finally, to quantify the precision of λ₁, λ₂, λ₃, and $\stackrel{‒}{\lambda }$ at each SNR and b value, the coefficient of variation (CV=100%*SD/mean) was calculated.

RESULTS

Diffusivities and Derived Indices: Muscle Tissue

The effects of SNR variation on λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA are shown for representative b-values in Figure 1. λ₁ was overestimated at low SNR for b=145 and 290 s/mm² and was underestimated at low SNR for all b-values of 435 s/mm² and greater (Figure 1A). The origin of this underestimation of λ₁ at moderate-to-high b and low SNR is found in the plot of S_X vs. b (Figure 2A), which reveals progressively higher baseline offsets and a decreased slope on the semi-log plot with decreasing SNR. In general, λ₂ and λ₃ were underestimated at low SNR levels for all b-values (Figures 1B and 1C, respectively); but note a single exception for λ₂ at b=145 s/mm², SNR=30. The b-value dependence of λ₂ and λ₃ differed qualitatively, however, in that λ₂ underestimation progressively increased as a function of b-value, while λ₃ underestimation was similar at b=145 and b=1160 s/mm² and lowest at b=435 s/mm². Qualitatively and quantitatively, the b and SNR dependences of $\stackrel{‒}{\lambda }$ were similar to that of λ₂ (Figure 1D). Figure 1E shows the dependence of the mean FA estimate on SNR and b. As b increased, the errors in FA estimation at low SNR changed from over- to underestimation.

Figure 1.

Dependence of the eigenvalues of D and their derived indices on b and SNR for ROI’s containing muscle tissue pixels only. Panels A-E show mean values for λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA, respectively (note changes in ordinate scales among the plots). As indicated in the legend to Panel A, data are shown for four representative b-values (145, 435, 725, and 1015 s/mm²) only. Error bars indicate the SD.

Figure 2.

Dependence of T₂-weighted signal intensity, S, and diffusion-weighted signal in the X-direction, S_X, on b-value and SNR. For clarity, data are shown for six representative muscle SNR values only. Panels A and B show the mean data for ROI’s containing muscle pixels only and adipose tissue pixels only, respectively. Lower SNR values reflect an increasing baseline component, resulting in positive offset at all b-values and lower apparent diffusivity (as indicated by reduced slope). Note differences in ordinate scales between panels A and B.

Figures 3A-E show the SNR_1%, SNR_5%, and SNR_10% values for λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA, respectively. In several plots, the SNR_5% and SNR_10% appear to be the same, but this is because the 5% criterion was met with SNR=5, the lowest value tested; hence the SNR_10% value may be artificially elevated by a small amount. The general pattern was for a U-shaped dependence of the minimum SNR requirements on b-value, although only the ascending portion of the curve was emphasized for λ₂ and $\stackrel{‒}{\lambda }$ . For λ₁ and λ₃, minima in SNR requirements occurred at either b=435 s/mm² or 580 s/mm². For 5% accuracy, the highest SNR requirement across all b-values was 21.5 and for 1% accuracy the highest SNR requirement was 47.1. For FA (Figure 3E), the SNR_1% requirements were generally higher than for λ_1-3, but for b=290-870 s/mm², were 48.0 or lower. For SNR_5% and this range of b-values, the largest SNR requirement was 23.2.

Figure 3.

Minimal SNR requirements for estimation of the eigenvalues of D and their derived indices for ROI’s containing muscle pixels only: dependence on b. Panels A-E show mean values for λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA, respectively (note change in ordinate scale in the FA plot). As indicated in the legend to Panel A, SNR_10%, SNR_5%, and SNR_1% are shown.

The precisions of the λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA estimates are shown in Figure 4. For all variables and all SNR’s, the CV was highest when using b=145 s/mm². For λ₁, λ₂, λ₃, and $\stackrel{‒}{\lambda }$ at b≥290 s/mm², the relative order of CV values changed as a function of SNR, but the CV’s were generally very low for SNR ≥40 and above (decreasing from 0.7-2.1% for an SNR=40 to 0.3% for $\stackrel{‒}{\lambda }$ , b=725 s/mm², and SNR=140). At SNR≥40 and for all variables, the lowest CV typically occurred at b=725 s/mm². For FA (Panel E), the CV was higher than for the other measures; for SNR=40 and b≥290 s/mm², the CV ranged from 4.5-5.4% and was lowest at b=580 s/mm².

Figure 4.

Dependence of CV for the eigenvalues of D and their derived indices on b and SNR for ROI’s containing muscle tissue pixels only. Panels A-E show the CV of λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA, respectively (note changes in ordinate scales among the plots). As indicated in the legend to Panel A, data are shown for four representative b-values (145, 435, 725, and 1015 s/mm²) only.

Diffusivities and Derived Indices: Adipose Tissue and Muscle/Adipose Partial Volume Artifacts

The assumptions concerning TR, TE, T₁, T₂, and ρ caused the measured adipose SNR to be 3.6-fold lower than the muscle SNR. Figure 5 shows the dependence of λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA in adipose tissue on b and SNR. With SNR ≥5, the mean value of λ₁ was always overestimated at the b-values tested; but at extremely low SNR (<5), a transition to underestimation similar to that of muscle λ₁ occurred. Figure 2B shows the plot of S_Xvs. b for adipose tissue and reveals the appearance of a noise floor similar to that in the muscle tissue. With one exception (for $\stackrel{‒}{\lambda }$ ), λ₂, λ₃, and $\stackrel{‒}{\lambda }$ were always underestimated at low SNR (Figures 5B, C, and D, respectively). FA was consistently overestimated and decreased monotonically with increasing SNR. Also of note is that for the b-values nearest those typically employed in muscle DT-MRI studies (435-580 s/mm²) and adipose SNR’s of ∼13-15 (muscle SNR’s of ∼50-60), the fat FA estimates (0.095-0.135) are well outside of the ±3% precision window that would be expected for muscle tissue from examination of Figures 1E, 3E, and 4E.

Figure 5.

Dependence of the eigenvalues of D and their derived indices on b-value and SNR for ROI’s containing adipose tissue pixels only. Panels A-E show mean values for λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA, respectively (note changes in ordinate scales among the plots). As indicated in the legend to Panel A, data are shown for four representative b-values (145, 435, 725, and 1015 s/mm²) only. Error bars indicate the SD.

Figure 6 shows the effects of partial volume artifacts on λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA. The mean values at the different f_M values range between the limiting values for adipose (f_M=0) and muscle (f_M=1). The parameter estimates do not depend linearly on f_M but rather are biased toward the muscle values. The data are shown only for b=580 s/mm², but a similar trend was observed for all b-values (with only the differences being in the limiting values at f_M=0 and f_M=1, as shown in Figures 1 and 5).

Figure 6.

Effects of muscle-adipose partial volume artifacts on the estimation of the eigenvalues of D and their derived indices. Panels A-E show mean values for λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA, respectively (note changes in ordinate scales among the plots). Mean values are given for b=580 s/mm² and six representative SNR values, as indicated in the legend to Panel A. Error bars indicate the SD.

Orientation of ε₁

When using 27-voxel ROIs, SNR≥20, and for all b-values tested, ε₁ was correctly oriented nearest the +X axis on all trials (Figure 7A). Figure 7B shows the f_E1X data for single-voxel data and indicates that b=435 s/mm² and SNR≥60, f_E1X=1; for the other b-values in the range 290-1015, SNR≥80 was required for f_E1X=1. Figure 7C shows the uncertainty of ε₁, as indicated by the SD of the distribution of α. The uncertainty decreases (precision improves) with SNR in a monotonic fashion; the b-value dependence is such that uncertainty is lowest in the range b=435-725 s/mm². Similar trends were noted for the single-voxel data, although the uncertainty was predictably higher (Figure 7D).

Figure 7.

Dependence of ε₁ orientation indices on b-value and SNR for 27-voxel ROI’s containing muscle tissue pixels only (A, C) or for single muscle voxels (B, D). Panels A and B show f_E1X at four representative b-values, indicated in the legend. Panels B and D show the SD of α across representative b- and SNR values.

Figures 8A and 8B show the SNR requirements for ±9° and ±4.5° uncertainty in the 27-voxel ROI, respectively, at four representative f_M values. For ±0.9° uncertainty, the SNR requirements were unrealistically high for human DT-MRI studies (>115) and so they have not been shown. For all f_M values, there was a U-shaped dependence of SNR_10% and SNR_5%, typically being centered on b=580 s/mm² and having a broad base of minimal or near-minimal values extending outwards by ±145 s/mm² or more. For b values within the range 435-725 s/mm² and f_M=5/9, SNR_5% averaged 40.9 and for f_M=4/9, SNR_5% averaged 50.3. Figures 8C and 8D show the single-voxel data, and reveal SNR_10% requirements of ∼54 in the range b=435-725 s/mm²; for completeness, the SNR_5% requirements have also been shown, although the SNR required for this level of precision is again unrealistically high.

Figure 8.

Minimum SNR requirements for estimating ε₁ orientation with 9 and 4.5° uncertainty in 27-voxel ROIs (Panels A and B, respectively) and with 9 and 4.5° uncertainty for single voxels (Panels C and D, respectively). Data are shown for four different f_M values (see legend to Panel A).

DISCUSSION

The study has examined the statistical errors associated with sampling D in muscle by examining the dependence of it and its derived indices on the b-value and SNR. It has also examined the statistical behavior of D in adipose tissue and in a full range of muscle-adipose partial volume artifacts. The results provide new information about the errors associated with estimating D in general and in muscle in particular, and have important implications for the design and interpretation of muscle DT-MRI studies.

Review and Interpretation of Key Findings

An important finding of this study was that in muscle, the statistical behavior of λ₁ in the low SNR level regime changes dramatically as b is progressively increased from 145 to 1160 s/mm². At low b-values, λ₁ is overestimated by ∼16%. Also, λ₂ and λ₃ are underestimated, respectively, by ∼6 and ∼33%. These phenomena have been reported and discussed previously and reflect the eigenvalue repulsion phenomenon of DT-MRI (26-29,31). At higher b-values and in the low SNR regime the underestimation of λ₂ and λ₃ persists, but the statistical behavior of the λ₁ changes to an underestimation of almost 40%. The origin of this behavior was revealed in Figure 2A, which shows the diffusion-weighted signal attenuation for a gradient direction (+X) that is collinear with ε₁. The use of magnitude images results in a baseline offset. For very low SNR levels, this offset is very large, but is unaccounted for when only two b-values are used in the estimation of D. The result is a reduced slope in the semi-log plot and an erroneously reduced apparent diffusivity. A similar effect is also seen in the adipose tissue component of the model tissue; however, the lower diffusivity in this tissue causes the effect to be quantitatively less important. This effect is consistent with the conclusions of previous reports (30,31).

We also used objective accuracy criteria (±10%, ±5%, and ±1%) to determine the minimal SNR requirements for estimating the components of D and their derived indices in muscle DT-MRI. The SNR requirements for λ₁, λ₂, λ₃, and $\stackrel{‒}{\lambda }$ all had slightly different dependences on b; but when considered collectively, the lowest SNR requirement was found at b=435 s/mm² and the overall pattern was U-shaped. For 5% accuracy and under all conditions tested, the SNR requirement for the T₂-weighted image was ∼20. For 1% accuracy, the SNR requirement for the T₂-weighted image was ∼40 for typical b-values. These findings confirm our prediction concerning the greater SNR requirement of muscle DT-MRI than of white matter DTMRI. Also noteworthy is that while we have here continued the convention of specifying the SNR requirement in terms of the T₂-weighted image, the observation that the λ₁ underestimation in both adipose and muscle tissues begins at similar diffusion-weighted image SNR levels (compare Figures 2A and B) suggests that the SNR requirements of DT-MRI might be better stated in terms of the lowest SNR of the diffusion-weighted images, not the T₂-weighted images.

As the present data and a previous report (31) show, the muscle λ₁, λ₂, and λ₃ estimates depend differently on b-value. These differences affect the statistical behavior of FA and its dependence on b, as well. The transition from λ₁ overestimation to λ₁ underestimation along increasing b-values, coupled with consistent λ₂ and λ₃ underestimation, causes FA to be overestimated in the low SNR regime at low b, relatively noise-stable at b=935 and 1015 s/mm², and underestimated at b=1160 s/mm². Conversely, in the isotropic adipose tissue, the overall tendency is for the eigenvalues to converge at higher b-value, and so FA approaches zero as SNR increases, regardless of the b-value. It is worth noting that the relative noise insensitivity of the muscle FA estimate at high b-values and the correspondingly low SNR_10%, SNR_5%, SNR_1% values should not be interpreted to mean that if FA is the primary parameter of interest, high b-values are best, because the relative noise insensitivity was achieved by severe underestimation of λ_1-3. Also, the CV for the FA estimates was about twofold higher than for the λ_1-3 estimates. The most likely explanation for this is error propagation resulting from the combination of multiple uncertainties.

The sensitivity of the orientation estimate of the diffusion ellipsoid was examined using f_E1X and α. Previously, data from ROIs such as those used in the present study have been used to estimate capillary orientation in models of the extravascular BOLD effects in muscle (17). For the 27-voxel ROIs, for all b-values tested, and SNR >20, f_E1X always equaled one; SNR≥60 (for b=435 s/mm²) or ≥80 (for other b-values in the range 290-1015 s/mm²) were required for f_E1X to equal one in single-voxel analysis. Also, we found predictable decreases in the uncertainty of α with increasing SNR. Among the b-values tested, the minimum SNR requirement for both 9° and 4.5 ° accuracy occurred at b=580 s/mm², having values of ∼20 and ∼40, respectively, for ROI analysis; for single-voxel analysis, SNR∼54 was required for ±9° uncertainty. Similar SNR requirements were obtained within the range ±145 s/mm² from this b-value.

An additional, and novel, aspect of this study was the examination of partial volume artifacts in ROI analyses and their effect on the estimation of D. As shown in Figure 6 and discussed in the Results, the eigenvalues and associated measures transitioned from the values for muscle to those for adipose as non-linear functions of f_M. An assumption of slow exchange (on the timescale of TE) between the muscle and adipose tissues was implicit in the averaging method used, and so on this basis a linear relationship might be expected; however the higher muscle SNR biases the mean diffusion characteristics toward the muscle tissue’s properties. Drawing a line of iso-diffusivity or iso-FA across the plots in Figure 6 indicates that for SNR>40, λ_1-3, $\stackrel{‒}{\lambda }$ , and FA met the 10% accuracy criteria when there was at least 7/9 muscle content in the region analyzed, and that the 5% accuracy criterion was met when there was at least 8/9 muscle content in the ROI. The uncertainty of α also depended non-linearly on f_M. This can be observed if a line of iso-SNR is drawn vertically across one of the plots of Figure 7. The SNR requirements for SD(α)=9° and 4.5° predictably increase with decreasing f_M, but for b=435-580 s/mm² and f_M values of 1/2 muscle and above take on reasonable values of ∼25 and ∼45, respectively.

The results and above discussion can be used to make specific recommendations with respect to b and SNR for ROI-based muscle DT-MRI studies. These are summarized in Table 2. Five additional points should be made. The first is that λ_1-3, $\stackrel{‒}{\lambda }$ , FA, f_E1X, and α were not calculated as the average of the individual pixel values, but rather after recalculation of D from the mean signal intensity values in the ROI. The latter approach reduces the eigenvalue bias by a factor of 1/N, where N is the number of pixels (28). Second, this study has considered only the statistical issues of sampling D in the presence of Rician-distributed noise. Other noise characteristics, such as uncorrected DC offsets, have not been considered. Third, greater SNR requirements may exist if additional processing steps, such as registration and B₀ inhomogeneity correction, are to be made. Fourth, the R matrix defined in Eq. 5 does not uniformly sample 3-dimensional space, but instead uses the “standard” encoding scheme of diffusion weighting along the X, Y, Z, XY, XZ, and YZ directions. This encoding scheme was chosen to facilitate comparisons with the neuronal DT-MRI literature, to allow the analysis made in Figure 2, and because it is the most commonly employed diffusion encoding scheme used so far in the muscle DT-MRI literature. Use of the uniform sampling scheme described by Jones et al, as opposed to the standard encoding scheme used here, was shown to reduce the SD of the Trace(D) and FA estimates in a water phantom by ∼30 and 55%, respectively (34). Finally, while ε₁ can be specified with reasonable accuracy for f_M values as low as 1/2 for reasonable SNR levels (∼45), the diffusivities cannot; so ROI specification must carefully exclude visible fat.

Table 2.

Summary of the accuracy and precision of estimating λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , and FA: dependence on b and SNR illustrated for three typical conditions. Calculation of D from mean signal intensity in an ROI of size 27 is assumed. For the accuracy values, the values were similar for the diffusivities and FA. The precisions of FA were lower than those of the diffusivities and are indicated in parentheses.

	Condition
Property	1	2	3
b (s/mm2)	435-725	435-725	435-725
fM	9/9	9/9	8/9
SNR	25	50	40
Accuracy (%)	5	1	5
Precision (%)	∼2 (∼5)	∼1 (∼3)	∼1.5 (∼4)

Implications for Muscle DT-MRI Studies

In physiological MRI studies of muscle, the SNR may change in several conditions. The first is in comparing diffusivity changes that follow exercise (36,37). During and after exercise, the muscle T₂ can increase by as much as 30% (38,39), increasing signal intensities in T₂-weighted images and increasing the SNR. This SNR change would move one’s position on the diffusivity-SNR curve to the right. If the SNR is initially low (∼20), then changes in λ₁ following exercise would be underestimated and changes in λ₂ and λ₃ would be overestimated, because of the statistical effect of noise. Another example is the edema corresponding to muscle damage. This water accumulation increases the overall T₂ of the tissue by increasing the volume fraction of the interstitial water compartment, which has a longer T₂ than intracellular water (40-42). The result is that in muscle diseases such as the inflammatory myopathies (10), after acute or prolonged ischemia (6,7), or after repeated lengthening contractions (43), T₂ and T₂-weighted signal intensity increase. If the SNR in the control measurements is not sufficient for the diffusivities and FA to have reached their asymptotic values, then any increases in λ₂ and λ₃ or decreases in FA — which might be interpreted as signs of membrane damage or edema — would again be confounded by the statistical effect of noise. However, if the SNR >40 then the 1% accuracy standard will likely have been met and any bias that would be introduced by SNR changes caused by the experimental intervention would be much smaller than any practically significant diffusivity changes. Furthermore, muscle DT-MRI studies should report the SNR so that the potential for these effects can be evaluated.

The findings of this study also have implications for muscle DT-MRI fiber tracking studies. First, the findings in Figures 7 and 8 suggest that stringent SNR requirements (≥60 for b=435 s/mm²; ≥80 for other values in the range b=290-1015 s/mm²) exist for tracking algorithms in which only the current voxel’s information is used to calculate the direction of fiber tract propagation. Related to this, the predictable decrease in α with increased ROI size suggests that methods for calculating the tract propagation direction that use information from multiple voxels, such as the tensorlines algorithm (44) or tri-linear interpolation, might improve the quality of muscle DT-MRI fiber tractography. Moreover, the predictable decrease in α with increasing f_M suggests that the application of these algorithms in muscle studies could be further improved by using a map of f_M (obtained by either setting a signal threshold in a T₁-weighted image, combining conventional and fat-saturated images, or Dixon imaging). This information could be used to weight the calculation of the fiber tracking step so that pixels with an f_M value below the certainty limit for the measured SNR are not considered in the weighting scheme. Finally, we have previously used fiber tracking data to calculate the pennation angle (1,4), which is the angle formed by the local tangent to the muscle fiber axis and the local tangent to the tissue of muscle fiber insertion (a tendon or aponeurosis). As the measurement is averaged over an increasing number of fiber tract points, two competing effects will occur: the measurement may be improved by including a larger number of points in the measurement, and the measurement may be biased by structural features such as fiber curvature (45). The number of points should be kept to the minimum required to estimate the true value, and will depend on the degree of fiber curvature and the actual SNR.

The final implication of these data for muscle DT-MRI studies is that SNR values greater than 40 (for measurements of λ₁, λ₂, λ₃, $\stackrel{‒}{\lambda }$ , or FA) or greater than 60 (for fiber tracking studies employing single-voxel algorithms) do not provide any statistical advantage in estimating D or any of its derived indices. This signal would be better used to increase spatial resolution, decrease the total imaging time, or to reduce gradient strength requirements, thereby decreasing the potential for eddy current artifacts.