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Keywords:

  • diffusion tensor imaging;
  • fractional anisotropy;
  • relative anisotropy;
  • SNR;
  • bootstrap;
  • Monte Carlo Simulations

Abstract

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS AND DISCUSSION
  6. Acknowledgements
  7. REFERENCES

Fractional anisotropy (FA) and relative anisotropy (RA) are the two most commonly used scalar measures of anisotropy in diffusion tensor (DT) MRI. While a few published studies have shown that FA has superior noise immunity relative to RA, no theoretical basis has been proposed to explain this behavior. In the current study, the diffusion tensor invariants were used to derive a simple analytical expression that directly relates RA and FA. An analysis based on that analytical expression demonstrated that the FA images have a higher signal-to-noise ratio (SNR) than RA for any value of tensor anisotropy RA or FA > 0. This theoretical behavior was verified using both Monte Carlo simulations and bootstrap analysis of DT-MRI data acquired in a spherical water phantom and normal human subjects. Magn Reson Med 51:413–417, 2004. © 2004 Wiley-Liss, Inc.

Diffusion tensor (DT) MRI is a sensitive technique that provides useful in vivo quantitative and qualitative maps of deep-tissue organization in both normal and pathological states (1, 2). A number of maps (e.g., orientation, coherence, and anisotropy) that reflect tissue organization can be derived from the eigenvalues and eigenvectors of the diffusion tensor (1–5). Relative anisotropy (RA) and fractional anisotropy (FA) are the most popular scalar indices used to visualize and quantify the diffusion anisotropy (3–6). These measures are rotationally invariant, and scale- and sorting-independent (7–9). A few published studies have shown that FA exhibits better noise immunity characteristics (10) and superior contrast-to-noise ratio (CNR) compared to RA maps (11, 12). For example, using Monte Carlo simulations, Papadakis et al. (10) demonstrated that FA maps diffusion anisotropy with the greatest detail and anisotropy signal-to-noise ratio (SNR). Similarly, Sorensen et al. (11) compared different anisotropy indices, using Monte Carlo simulations at different SNR levels, and concluded that FA provided the highest CNR between gray and white matter. By a region-of-interest (ROI) analysis, Alexander et al. (12) demonstrated that in general FA provides superior discrimination between gray and white matter, and between different white matter tracts, compared to RA. However, to the best of our knowledge, no theoretical basis has been proposed that rationalizes these observations in terms of the relative sensitivity and contrast of these two measures. Therefore, the main purpose of the current study was to investigate the relationship between RA and FA and provide a theoretical basis on which to relate the noise characteristics of these two measures. The contrast between regions with different anisotropy depends on the working SNR level and the selection of the anisotropy metric (5, 13). To better understand the sensitivity of these theoretically-related indices to the measurement noise, Monte Carlo simulations and a bootstrap analysis (a nonparametric statistical technique) were performed on DT-MRI data acquired in a spherical water phantom and normal human brains. These studies were carried out at different SNR levels on data acquired using rotationally invariant and icosahedral tensor encoding schemes.

THEORY

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS AND DISCUSSION
  6. Acknowledgements
  7. REFERENCES

Tensor Invariants, Mean Diffusivity, RA, and FA

The three rotationally-independent principal tensor invariants are defined as I1= tr(D) = 3 μ(D), I3 = det(D), and I2 = I3tr(D1), where tr and det are the trace and determinant of the matrix, respectively (14). The Frobenius or tensor norm, ∥D∥, is considered to be a fourth invariant and is defined as the root of I4= tr(D2) = Imath image – 2I2 (14). These invariants can be expressed either directly in terms of the diffusion tensor elements (8) or the eigenvalues (λ1, λ2, and λ3):

  • equation image(1)
  • equation image(2)
  • equation image(3)
  • equation image(4)

The commonly used tensor anisotropy indices, RA (5) (also related to the aspherism index (6, 15)), Aσ (3), Afiber (8), Asd (13)), and FA (3), are related to the mean diffusivity, μ(λ) = I1/3, eigenvalue standard deviation (SD), σ(λ), the tensor norm (∥D∥), and the invariants (14):

  • equation image(5)
  • equation image(6)

Note that both RA and FA are normalized so that 0 ≤ RA ≤ 1 and 0 ≤ FA ≤ 1. The maximum anisotropy value corresponds to the case RA = FA = 1, which is attained when λ1 > λ2 = λ3.

Relationships Among RA, FA, Anisotropy Sensitivity, SNR, and CNR

The relationship between RA and FA can now be expressed explicitly using the previously defined invariants as:

  • equation image(7)

Equation [7] can be used to derive a closed-form relationship between σ(RA), σ(FA) and the anisotropy SNR (10), SNR(FA) = FA/σ(FA) and SNR(RA) = RA/σ(RA):

  • equation image(8)
  • equation image(9)

The derivative of FA with respect to RA was used as a measure of anisotropy sensitivity to noise, dFA/dRA = σ(FA)/σ(RA), by Ulug and Van Zijl (8) and Armitage and Bastin (13). The SNR(FA) and SNR(RA) follow the definition adopted by Papadakis et al. (10) for comparing the noise immunity of various anisotropy measures using Monte Carlo simulations. The CNR between two regions with signal intensities x and y in the DTI images can be expressed as (11, 12, 16):

  • equation image(10)

Note that if σx ∼ σy = σ and |x – y| > σ, the CNR will be proportional to the intrinsic SNR difference between the two tissue regions. An anisotropy measure that gives the largest SNR difference is clearly desirable. If |x – y| ∼ σ, the CNR would be determined by the ratio ∼ |x – y| /σ. In this case, a gain in SNR may not necessarily improve CNR.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS AND DISCUSSION
  6. Acknowledgements
  7. REFERENCES

DT-MRI Data Acquisition

DT-MRI data were acquired in spherical water phantoms and consenting normal human subjects using a diffusion-sensitized, dual spin-echo-prepared, echo-planar imaging (EPI) sequence that utilized ramp sampling and spectral selective pulses for fat suppression. The dual spin-echo option helps reduce eddy current-induced geometric distortions (Ref. 17 and references therein). For the water phantom acquisition the parameters were TR = 6000 ms, TE = 67 ms, FOV = 240 mm, slice thickness = 3 mm, b-factor = 500 s mm–2. A multifaceted and rotationally-invariant icosahedral encoding scheme (17) with 21 encoding directions (Ne) was used to acquire the diffusion-weighted (DW) images. This encoding scheme provides three tensor estimates (principal icosahedron vertices (Icosa6), principal edge bisectors (Icosa15), and Icosa21 = Icosa6 + Icosa15 vertices) at three rotationally invariant and uniformly distributed sets using the same DW data (18). Thus, it is possible to extract DTI data from this encoding scheme with different levels of SNR.

DT-MRI Data Acquisition for Bootstrap Analysis

Multisection DT-MRI data sets were acquired for a bootstrap statistical analysis on four normal subjects. To reduce artifacts arising from CSF pulsation, the DT-MRI data were acquired using cardiac pulse gating. On each subject, the sequence acquired two slices with Nref = 19 images at b ∼ 0 (ref pool) and 21 DW images using the Icosa21 encoding scheme (DW pool). The image acquisition parameters were: TR = 3 RR intervals, TE ∼ 84 ms, and slice thickness = 3 mm with a gap of 5 mm. The data in the ref and DW pools were coregistered and distortion-corrected with the AIR package (Ref. 17 and references therein). The ref and DW image pool measurements were repeated sequentially with Nrepeats = 12, and the total number of images acquired for each slice was NT = (Nref+ Ne) Nrepeats = 480. The computer-intensive bootstrap analysis was performed by randomly resampling the original data pools (ref and DW) with a replication number NB ∼ 5000 at different SNR levels. A full description of the application of the bootstrap procedure to DT-MRI is beyond the scope of this work, but details can be found elsewhere (14, 19).

Monte Carlo Simulations

DT-MRI Monte Carlo simulations using a numerical phantom were also carried out to validate the simple theoretical relationship between SNR(FA) and SNR(RA) (17). Briefly, the phantom has a highly anisotropic 256 × 256 region with eigenvalues λ = [1.60, 0.40, 0.40] × 10–3 mm2 s–1, (RA = 1/2, FA = 1/√2), a planar anisotropy 512 × 128 region with λ = [1.0, 1.0, 0.40] × 10–3 mm2 s–1 (RA = 1/4, FA = 1/√6), and a 128 × 256 isotropic region with λ = [1.60, 1.60, 1.60] × 10–3 mm2 s–1 (RA = FA = 0). The Monte Carlo simulation procedure used and the DT-MRI data processing are described elsewhere (5, 9, 13, 17, 18).

RESULTS AND DISCUSSION

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS AND DISCUSSION
  6. Acknowledgements
  7. REFERENCES

The results of the bootstrap analysis are summarized in Fig. 1. The mean RA and FA maps obtained from the 5000 replicated bootstrap analyses are shown in Fig. 1a and b. The superior image quality of FA relative to RA can be easily appreciated from these images. The SD and SNR maps derived for RA and FA from the bootstrap analysis are shown in Fig. 1d, e, g, and h. The higher SNR of the FA maps relative to the RA maps are evident in these figures. These maps also clearly indicate that the FA maps have higher CNR between white matter tracks and neighboring CSF and gray matter, compared to the RA maps. The scatterplots derived from the bootstrap analysis, along with the theoretical curves derived from Eqs. [7][9], are shown in Fig. 1c, f, and i. Note that the scatterplots are indistinguishable from the theoretical curves, demonstrating the excellent agreement between the theoretical relationship and experimental data. The maximum RA value in human brain is 0.6, and therefore the bootstrap results are not shown beyond this value of RA.

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Figure 1. Summary of the bootstrap results from a normal human brain. The results shown in this figure correspond to the selection of Nref = 10 from the b = 0 pool, and Nd = 10, Ne = 6 from the DW data pool. The maps in a, b, d, e, g, and h show the mean RA and FA, the σ(RA), the σ(FA), the SNR(RA), and the SNR(FA), respectively. The plots in c, f, and i show the scatterplots of the experimental results with the theoretical curves predicted for FA(RA), σ(FA)/σ(RA), and SNR(FA)/SNR(RA).

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Figure 2 shows a few representative DT-MRI measurements of SNR(FA)/SNR(RA) in the central region of a water phantom. The water phantom serves as a useful and sensitive model for isotropic diffusion in edema or CSF. Figure 3 shows some typical results of the DT-MRI Monte Carlo simulation for different anisotropy (spherical, planar, and cylindrical tensors) and icosahedral encoding schemes (Icosa6, Icosa15, and Icosa21) derived from the major Icosa21 scheme. It is important to point out that all three schemes provide unbiased estimation of FA and RA values and differ only in terms of their SNR. Of the three schemes, the Icosa21 and Icosa6 schemes provide the highest and lowest SNRs, respectively. The spherical tensor case in Fig. 3a and the water phantom in Fig. 2 are used to study the artifactual anisotropy that results from the measurement noise at finite SNR. Since 0 ≤ RA ≤ 1, the SNR(FA)/SNR(RA) formulas (Eqs. [7][9]) predict that the SNR in FA is always higher than in RA for anisotropic regions, consistent with the data shown in Fig. 1. This anisotropy can be structural or artifactual, due to SNR sensitivity. The maximum SNR(FA) advantage over SNR(RA) is threefold at the highest theoretical anisotropy value of RA = FA = 1. In practice, the largest computed RA values in human brain, as demonstrated in Fig. 1, are about 0.6 and hence the SNR(FA)/SNR(RA) ratio will be about 1.7. The theoretical asymptotic relationship between SNR(FA) and SNR(RA) is independent of the DT-MRI acquisition parameters, such as the reference (b ∼ 0) image SNR (SNR0), tensor encoding scheme, and the b-factor (8, 18, 19). It is worth pointing out that the Monte Carlo simulations, the water phantom measurements, and the brain measurements were performed at finite SNR0 ∼ 50. At finite SNR values, the anisotropy value obtained on a spherical phantom is always overestimated (5, 13, 20) and the measured SNR ratio, SNR(FA)/SNR(RA), is larger than unity and therefore results in a better contrast between different tissue types on the FA maps. The results from the current theoretical analysis confirm previous studies (e.g.,10 that used Monte Carlo simulations to compare different anisotropy measures. Equation [9] can be used to explain and confirm the tabulated Monte Carlo simulation results conducted at a broader range of anisotropy values in Table 1 of Ref. 10. Similar simulations are shown in Fig. 3 and are validated on the water phantom in Fig. 2. The current results provide a theoretical foundation for recent findings (11, 12), from ROI analyses on brain measurements and Monte Carlo simulations, that FA noise immunity is superior to RA. The theoretical sensitivity relation σ(FA)/σ(RA) can also be used directly to explain the numerical results reported by Armitage and Bastin (13) and Ulug and Van Zijl (8).

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Figure 2. Summary of SNR(FA)/SNR(RA) = [μ(FA)/σ(FA)]/[μ(RA)/σ(RA)] on the central region of a spherical water phantom for different icosahedral encoding schemes (Icosa6, Icosa15, and Icosa21), with number of encoding directions (Ne) = 6, 15, and 21, respectively. Data were acquired using b-factor = 500 s mm–2 and the Icosa21 encoding scheme. Notice that the ratios SNR(FA)/SNR(RA) follow those predicted by the theory for the isotropic case (RA ∼ 0), which are also shown in Fig. 1c, f, and i. The deviation from the theoretical curve is attributed to residual artifactual anisotropy, which decreases as the number of DW images increases.

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thumbnail image

Figure 3. Results of typical Monte Carlo simulations of SNR(FA)/SNR(RA) for different tensor shapes ((a) spherical, (b) planar, and (c) cylindrical) for RA = 0, 0.25, and 0.50, respectively, and using different icosahedral encoding schemes derived from the Icosa21 scheme. The number of encoding directions (Ne = 6, 15, 21) corresponds to Icosa6, Icosa15, and Icosa21, respectively. The simulation parameters are b-factor = 1000 s mm–2 and SNR0 = 50. Notice that the ratios SNR(FA)/SNR(RA) follow those predicted by the theory, which are also shown in Fig. 1c, f, and i, and Fig. 2 for the isotropic case. The deviation from the theoretical curve is attributed to dependence on the encoding scheme and the b-factor noise sensitivity. The asymptotic SNR(FA)/SNR(RA) values predicted by Eq. [9] are 1, 1.125, and 1.5, respectively.

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In conclusion, we have shown on theoretical grounds that the noise characteristics of FA are superior to RA. This theoretical result was validated using DT-MRI Monte Carlo simulations, water phantom measurements, and bootstrap DT-MRI analyses on normal subjects.

Acknowledgements

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS AND DISCUSSION
  6. Acknowledgements
  7. REFERENCES

This work was funded in part by the Radiology Department of the University of Texas, Houston, and NIH grants to P.A.N. (NIBIB R01 EB02095) and A.L.A (NIBIB RO1 EB002012). The authors thank Marci A. Harris for editorial assistance.

REFERENCES

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS AND DISCUSSION
  6. Acknowledgements
  7. REFERENCES