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Keywords:

  • magnetic resonance imaging;
  • diffusion tensor encoding;
  • fractional anisotropy;
  • relative anisotropy;
  • mean diffusivity;
  • icosahedral encoding;
  • Monte Carlo simulations

Abstract

  1. Top of page
  2. Abstract
  3. THEORY
  4. METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

Diffusion tensor MRI (DT-MRI) is a promising modality for in vivo mapping of the organization of deep tissues. The most commonly used DT-MRI invariant maps are the mean diffusivity, μ(D), relative anisotropy (RA), and fractional anisotropy (FA). Because of the computational burden, anisotropy maps are generally computed offline. The availability of a simple procedure to compute RA, FA, and μ(D) online would make DT-MRI more useful in clinical applications that require immediate feedback. In this study, analytical expressions that relate the commonly used tensor anisotropy measures obtained from the decoded and diagonalized DT with those obtained from the first and second moments of the measured diffusion-weighted (DW) data are derived. Specifically, it is shown that for the principal icosahedron encoding scheme, RA is related to the mean and standard deviation (SD) of the DW measurements that can be computed online. Since FA is commonly used as an anisotropy measure, an analytical expression relating RA and FA was derived from the tensor invariants. These results were validated using both Monte Carlo simulations and high-resolution, normal whole-brain DT-MRI measurements acquired with different b-factors, encoding schemes, and signal-to-noise ratio (SNR) levels. The bias introduced by the rotationally variant encoding schemes into the diffusion measures is also investigated. Magn Reson Med 50:589–598, 2003. © 2003 Wiley-Liss, Inc.

Diffusion tensor MRI (DT-MRI) has become an increasingly important modality for understanding the organization of normal brain structures (1–4) and the evolution of neurological and psychiatric disorders (1, 5–8). Currently, there are different methods for acquiring, processing, and modeling the measured DT-MRI data (1, 9–14). However, there are many technical and scientific challenges limiting the interpretation, validation, and assignment of the true contributors to the DT-MRI signal in complex biological systems (1, 7–17). In general, the DT-MRI derived and rotationally invariant maps, such as μ(D), RA, and FA (6, 18, 19), are computed offline because of the intense computational nature of the analysis. The availability of online DW procedures to compute these measures would extend the utility of DT-MRI to clinical applications for acute diseases, where immediate feedback to the attending clinicians is important.

It was recently shown that useful information can be gained by applying spatially independent component analysis (offline and computationally intensive) and higher-moment statistics to the diffusion data (12). Additional studies (20) have suggested that anisotropy measures can be used to investigate the complex fiber structures within a voxel without certain model assumptions about the signal sources. The relationship between DW-based invariants and those obtained from the single tensor model (when it is an operationally acceptable working model) has not been formally explored. The influence of bias introduced by the encoding scheme and the SNR has also not been addressed.

There is considerable evidence that icosahedral encoding sets are the least biased and the most rotationally invariant sets (9, 21–26) suitable for acquiring DW data. This work shows that by employing the uniformly distributed principal icosahedron sampling scheme, a diffusion anisotropy measurement analogous to FA can be directly obtained from the DW data. The effects of SNR, encoding scheme, and diffusion sensitization on the accuracy of the method are also investigated using Monte Carlo simulations and normal brain DT-MRI measurements.

THEORY

  1. Top of page
  2. Abstract
  3. THEORY
  4. METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

Before the algorithm and the requirements to obtain the rotationally invariant μ(D), RA, and FA maps directly from the DW data are presented, the basic mathematical underpinnings of the DT-MRI tensor estimation and encoding theory, using matrix algebra, are briefly summarized (1, 24–27).

Self-DT Encoding Theory

Since the DT is symmetric, a minimum of six noncollinear encoding directions are needed to obtain the six independent elements of the DT, D, represented as a 3 × 3 matrix (1–3):

  • equation image(1)

The unique tensor elements can be represented by the column vector (24):

  • equation image(2)

In the absence of internal gradients, external gradients, and restrictions (1, 9, 17–19), the single tensor model that relates the unknown tensor, D, to the encoding diffusion unit vector, equation image(n) = [gx(n) gy(n) gz(n)], the diffusion weighting b-factor, b, the reference image pixel signal, S(0), and the measured diffusion weighted signal, S(n), can be expressed for n = 1…Ne as

  • equation image(3A)

where Ne is the number of encoding directions and Dm(n) = equation image(n)tDequation image(n) is a scalar quantity that can be viewed as the projection of the DT onto the measurement encoding unit vector equation image. To enhance the SNR and reduce phase fluctuations, the reference and DW images are repeated and magnitude averaged Nref and Nd times, respectively, for each of the Ne directions. Thus, the total number of images acquired is NT= Nref+ NeNd. The column vector, equation image, is formed from the measured data, with components

  • equation image(3B)

For symmetric trapezoidal, unipolar diffusion gradient pulses of strength gd, rise time ε, duration δ, and separation Δ, the diffusion b-factor is given by (28)

  • equation image(4)

where γ is the gyromagnetic ratio. Note that Dm(n) can also be expressed as a product Dm(n) = equation image(n)t · equation image, where

  • equation image(5)

The relationship between Y and d can be expressed in matrix notation as

  • equation image(6)

where equation image is the measurement noise vector. The design encoding matrix, H, is formed from the row vectors equation image(n)t. The least-squares optimization and the singular value decomposition methods (24–27) can now be used to estimate the unique DT elements, vector, equation image, where the superscript “–1” refers to matrix pseudo-inversion:

  • equation image(7)

Equations [ 6] and [7] can be used to compute the covariance matrix of the estimated diffusivities in terms of the encoding design matrix (H) and the measured noise matrix covariance (17, 24–27). A useful, but crude, diagnostic figure-of-merit to characterize the coupling between the noise in the measured data and the estimated diffusivities is the condition number of H, defined as the ratio of the maximum to the minimum eigenvalues of H or κ(H) = max (eigvals (H))/min (eigvals (H)) (24–26, 29). Uniformly distributed and rotationally invariant encoding schemes are characterized by an icosahedral condition number of 1.5811. A condition number smaller or larger than 1.5811 indicates a rotationally variant and nonuniformly distributed set (22, 26).

Tensor Invariants and the Estimation of the μ(D) and Tensor Anisotropy Measures

The following quantities are defined: a1= Dxx2+ Dyy2+ Dzz2, a2= DxxDyy+ DxxDzz+ DyyDzz and a3= Dxy2+ Dxz2+ Dyz2. The DT elements, equation image, can be used to estimate the magnitude, anisotropy, and orientation of the local DT. The three rotationally independent principal tensor invariants are defined as (2) I1= tr(D) = 3μ(D), I3= det(D) and I2= I3tr(D1) = a2– a3 (tr and det are the trace and determinant, respectively, of the matrix). 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. The three principal invariants can be used to compute the eigenvalues, the orthonormal eigenvectors of the tensor, and the different tensor anisotropy measures (30). The mean diffusivity coefficient is defined as μ(D) = I1/3. In addition, the commonly used tensor anisotropy indices, RA and FA, are related to the mean (μ), standard deviation (σ), the tensor norm (∥D∥), and the invariants (30, 31):

  • equation image(8A)
  • equation image(8B)

Relationship Between RA and FA

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

  • equation image(9)

The anisotropy measure RA, which is also proportional to the axial anisotropy, Aσ (6), has been shown to be a fundamental measure of aspherism in the trilinear coordinate system (30).

Analytical Encoding Method and the Relationship Between Tensor Encoding Schemes

The analytical encoding method (24–26) was used to investigate the relationship between different encoding schemes. The method constructs the simplest encoding matrix for Ne = 6 from cyclic permutations of the normalized generating vector basis equation image(u) = 1/ equation imageu ±1 0]t:

  • equation image(10)

Using Eq. [ 5], the H matrix can be constructed as a function of the continuous variable u:

  • equation image(11)

Equations [ 6] and [11] can be used to obtain μ(D) by taking the expectation value of the column sum or

  • equation image(12)

This property implies that the generating set, Em(u), can provide an unbiased estimate of μ(D) directly from the measured DW and reference data for any value of u. This analytical and general encoding matrix can be applied to generate three familiar cases. For u = 1 one can obtain the cube edge bisectors set (31); the condition number of this scheme = 2.00. The principal icosahedron encoding scheme is obtained for ui cos a6 = τ = 2 cos(π/5) = (equation image + 1)/2 and u = 1/τ = (equation image − 1)/2 with condition number = equation image/2 ≅ 1.581 (22, 26). The minimum condition number set is obtained with u = umc = equation image)/2 and a condition number of equation image/2 ≅ 1.323 (22, 29). The principal icosahedron scheme is the only uniformly distributed and rotationally invariant set with Ne= 6 (21–26).

Relationship Between the DW and DT Variance

Define the first moment of the measured data vector equation image and using Eqs. [ 6] and [11]:

  • equation image(13)

where Sumc stands for the column summation of the matrix H. For a family of balanced and encoding sets generated using the vector basis Eq. [ 10], the expectation value of all measurements is related to the first tensor invariant as m1 = μ(D) = I1/3. Using Eqs. [6] and [11], and the fact that tr(AB) = tr(BA), the second moment of the DW measurements can be defined as

  • equation image(14)

Many dimensionless and scale-independent anisotropy measures can be defined from the moments of the DW data, which are used to obtain the vector equation image (Eq. [ 3]):

  • equation image(15A)

and

  • equation image(15B)

These DW-based anisotropy measures are related as Aniso2 = equation image. Notice that m2(uicosa) = 1/15(3a1+ 2a2+ 3a3), and m2(u) – m2(uicosa) = 2/15(a1– a2 2a3)f(u), where f(u) = (u2– u – 1)(u2+ u + 1)/(u2+ 1)2. The zeros of f(u) occur at the golden ratio values of u = ±(±equation image − 1)/2. The relationship between Aniso1(equation image) and RA can be reduced following some algebraic and symbolic reductions:

  • equation image(16)

The value of Aniso1((equation image), u)/RA(D) is independent of u for only u = uicosa6. Thus, using the isotropically distributed principal icosahedron encoding matrix or Em(τ), the anisotropy measure obtained from the first and second moments of the DW measurements is related to the RA measure obtained assuming the single tensor model.

Relationship Between FA and the Aniso1(Y)

Using the closed-form relation between RA and FA, and Eqs. [ 9] and [16] relating RA with Aniso1(equation image), a fractional anisotropy measure from the DW data can be defined as

  • equation image(17)

This simple equation provides an exact and direct nonmodel algorithm to estimate FA from the DW data using the principal icosahedron encoding scheme. Since Aniso1 and Aniso2 are related to the mean and SD of the DW measurements, this expression can be computed without the need to decode or diagonalize the tensor.

METHODS

  1. Top of page
  2. Abstract
  3. THEORY
  4. METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

MR Acquisition

DW data from four healthy volunteers were acquired on a 1.5 T General Electric echo speed CNV MRI scanner using a standard transmit-receive RF birdcage head coil. The gradient system is capable of providing maximum gradient amplitude of 40 mT/m on all axes with a slew rate of 120 Tesla/m/s. All human studies were conducted in compliance with the regulations of the University of Texas at Houston Committee for Protection of Human Subjects.

Diffusion MRI Acquisition

All images were acquired with a diffusion-sensitized, spin-echo-prepared, single-shot echo-planar imaging (SE-EPI) sequence with spectral-selective RF pulses for fat suppression (1). The GE product sequence was modified to acquire the DT-MRI encoded data. Contiguous 3-mm-thick slices covering the whole brain were acquired in a sequential mode with a field of view (FOV) of 240 × 240 mm2. The echo time (TE) ranged from 72 to 80 ms, with a repetition time (TR) of 4–8 s. To minimize the distortion and image artifacts, the sequence utilized partial ky acquisition of 80/128 views with ramp sampling and an echo spacing of 642 μ. The data were zero-filled to attain an image matrix of 256 × 256. On each volunteer, four full normal brain data sets were acquired at two different levels of diffusion weighting: b = 1000 s mm–2 and 2000 s mm–2 at NEX = 4 and 8, respectively. The selection of the slice thickness and NEX gave an acceptable SNR for both the b = 0 and DW images in <15 min.

DT Encoding Schemes

In these studies a tessellated version of the principal icosahedron (21–26) DT encoding scheme was used (Icosa21; Table 1 and Fig. 1). Many optimal encoding sets can be derived from this scheme and its dual, the dodecahedron (24). Thus, one DW-MRI data set can be used to provide estimates from different encoding schemes without the need to acquire different data sets. This encoding scheme contains the three orthogonal directions (x, y, and z, or xyz for simplicity). All of the encoding schemes and condition numbers are listed in Table 1. The three rotationally invariant icosahedra-based sets derived from this scheme (Icosa6, Icosa15, and Icosa21) have a condition number κicosa of 1.5811 (Fig. 1 and Table 1). Note that the Icosa15 and Icosa21 encoding sets also satisfy the theoretical result in Eq. [ 16], which is derived using the Icosa6. Additionally, the data set was used to obtain many perturbed icosahedral sets (in particular, the one that was formed by replacing one vertex of the Icosa6 with the (1,0,0) direction (Ne = 6, condition number κ = 2.63)) and suboptimal yet balanced subsets S9 = Icosa6 + xyz (Ne = 9, κ = 1.37) and S12 = Icosa15-xyz (Ne = 12, κ = 2.00). A detailed discussion of the design properties and potential applications of this encoding scheme is beyond the scope of this study and will be presented in a separate article.

Table 1. Listing of the Balanced and Normalized Icosa21 Tensor Encoding Schemes
 123456789101112131415161718192021
  1. Define τ = 2 cos(π/5); A = 1/√(1 + τ2); B = τ/√(1 + τ2); C = (τ − 1)/2; D = τ/2.

  2. The order presented here is arbitrary and should be application dependent.

  3. The orthogonal xyz set is # [21, 4, 1]; Icosa6 = # [2, 3, 13, 14, 19, 20]; Icosa15 = [Icosa21 − Icos16]; perturbed − Icosa6 = #[2, 3, 13, 14, 19, 21]; S9 = Icosa6 + xyz; S12 = Icosa15 − xyz. Condition number κ(Icosa6, 15, 21) ∼ 1.58; κ(perturbed Icosa6) ∼ 2.6; κ(S9) ∼ 1.37; κ(S12) = 2.0.

  4. Note the 6 sets: xyz, Icosa6, S9, S12, Icosa15, and the full Icosa21 satisfy the requirement that Sumc(H) ∗ equation image/Ne = μ(D). The μ(D) map can be estimated using the DW data in 6 different ways.

gx0000CC−CC−.5.5.5.5AA−DDD−D−B−B−1
gy0A−A1−DD−D−DC−CCC−BB.5−.5.5−.5000
gz1−B−B0−.5−.5−.5.5DD−DD00CC−C−CA−A0
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Figure 1. Schematic representation of the Icosa21 (Table 1) and the subencoding sets: Icosa6, perturbed Icosa6, S9 = Icosa6 + xyz, S12 = Icosa15 – xyz, Icosa15 = Icosa21 – Icosa6, and Icosa21.

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DW and DT-MRI Data Postprocessing

The magnitude-averaged images were registered and distortion-corrected using the Automated Image Registration (AIR) package (12, 32). The DW image for each slice, with the least distortion based on a water phantom calibration exam acquired with identical image parameters, was used as the reference. The distortion-corrected images were then decoded using a least-squares singular value decomposition approach to estimate the DT elements (24–27). The DW-based μ(D) and anisotropy measures were computed following the procedures described in the Theory section. The tensor diagonalization and anisotropy maps were computed based on a thresholded intensity erosion-dilation and positive definite mask (30). To minimize pixel broadening, no spatial filtering was used. The DT and DW-MRI analyses, image processing, Monte Carlo simulations (described in the next section), and symbolic computations described in the Theory section were performed using an in-house-developed DT-MRI design and analysis toolbox under MATLAB (v6.1; Mathworks Inc., Natick, MA).

Monte Carlo Simulations

In order to understand some of the experimental results (see below), and to test the theoretical relationship between the DT and DW mean diffusivity and FA measures, Monte Carlo simulations (18, 24, 33) were carried out using a DT-MRI simulation module. The DT-MRI acquisition and processing described in the Theory section was implemented using the numerical phantom shown in Fig. 2. The 512 × 512 phantom was designed to simulate four different tissue types that were assigned different eigenvalues (λ), orientations, and reference intensities (S(0)). The phantom contains highly axial anisotropic white matter (central 256 × 256 region: λ = [1.60, 0.40, 0.40] × 10–3 mm2 s–1, S(0) = 4000, FA = 0.707); planar anisotropic white matter (upper 512 × 128 region: λ = [1.0, 1.0, 0.40] × 10–3 mm2 s–1, S(0) = 4000, FA = 0.408); lower 512 × 128 region: λ = [1.0, 0.40, 1.0] × 10–3 mm2 s–1, S(0) = 4000, FA = 0.408); CSF (left 128 × 256 region: λ = [3.20, 3.20, 3.20] × 10–3 mm2 s–1, S(0) = 10000, FA = 0); and edema (right 128 × 256 region: λ = [1.60, 1.60, 1.60] × 10–3 mm2 s–1, S(0) = 8000, FA = 0). Diffusion data on this synthetic phantom were generated according to the formalism in the theory, and by adding complex normally distributed noise at some prescribed SNR level, encoding scheme, and b-factor. A detailed description of the Monte Carlo procedure was given by Bastin et al. (33). Note that the SNR was selected based on the non-DW signal, S(0), in the white matter regions. The Monte Carlo replication number was equivalent to the number of pixels in each region. The processing was performed on the magnitude data as described above, and accelerated using the analytical tensor diagonalization method (30).

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Figure 2. A schematic map of the numerical DT-MRI phantom used in the Monte Carlo simulations. The phantom contains five different regions with different diffusion attributes, which are listed in the Monte Carlo Simulations section of Methods.

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RESULTS

  1. Top of page
  2. Abstract
  3. THEORY
  4. METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

Experimental Validation

Figure 3 shows the μ(D) and FA maps that were computed with and without the tensor model, as well as the corresponding absolute difference maps. These maps were generated for different encoding schemes with different b-factors, NEX, and SNR values (data not shown). For the sake of clarity, a few maps are shown at higher magnification in Fig. 4 for the Icosa21 scheme with NEX = 8 and b = 1000 s mm–2. Figure 5 shows the quantitative histogram plot of FA obtained with Icosa6, -15, and -21. As can be seen in this figure, the estimated FA values decrease as the number of encoding directions increases for all the three encoding schemes. Since the SNR increases with the number of encoding directions, this observation is consistent with the notion that noise results in an artifactual increase in the anisotropy value. Figures 3–5 indicate that at Ne= 6, only the Icosa6 resulted in a rotationally invariant anisotropy estimation, as indicated by the absolute difference maps |FA(DT) – FA(DW)|. This behavior is consistent with the theoretical formalism presented here. However, this is not the case for the perturbed Icosa6, as indicated by absolute μ(D) difference maps, |μ(DT) – μ(DW)|. The S9 and S12 behavior is different from that of the Icosa15 and Icosa21. The absolute μ(D) difference maps computed using the S9, S12, Icosa15, and Icosa21 indicate that these subsets can be used to obtain the μ(D) directly, although the S9 and S12 are rotationally variant. Using the data acquired with higher angular resolutions (Icosa15 and Icosa21), the difference maps |FA(DT) – FA(DW)| indicate that the largest differences occur in the gray matter and CSF regions, and the smallest values arise in the white matter regions. These observations hold for all four subjects at both low and high b-factors and for different SNR levels. The S9 and S12 schemes (as characterized by the condition numbers of 1.37 and 2.00, respectively) are rotationally variant and are expected to result in a biased estimate of the anisotropy (22, 26). However, these two schemes are still able to provide unbiased estimates of the μ(D).

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Figure 3. Experimental validation of the theoretical relationship between the DW and DT-based μ(D), and FA maps for various encoding schemes (shown in Table 1). The face triangularization of the encoding sets is shown in Fig. 1. a: μ(DT) (mm2s–1). b: μ(DW) (mm2s–1). c: |μ(DT) – μ(DW)| (mm2s–1). d: FA(DT). e: FA(DW). f: |FA(DT) – FA(DW). Data were acquired with NEX = 8 and b-factor = 1000 s mm–2.

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Figure 4. Magnified images for the Icosa21 case illustrated in Fig. 3 with NEX = 8 and b-factor = 1000 s mm–2. a: FA(DT). b: FA(DW). c: |FA(DT) – FA(DW)|. d: |FA(DT) – FA(DW)|/FA(DT). Note that d scales the absolute FA difference map by the FA(DT) to highlight regions of low anisotropy (gray matter and CSF).

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Figure 5. A FA histogram plot at different encoding schemes using the FA(DT) and FA(DW) methods, and the data set in Figs. 3 and 4. Note that 1) the results for the Icosa6 are identical, 2) the anisotropy estimation is dependent on SNR, and 3) at high SNR the DW and DT anisotropy FA estimates are identical.

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Monte Carlo Simulations

As described above, the experimental results in human brain are consistent with the theoretical predictions. However, it is possible that the finite values seen in the difference FA maps (especially around the low anisotropic structures of GM and CSF) could arise due to the errors in the estimation of FA values. To address this issue, we performed simulations using a synthetic phantom (Fig. 2). The simulations were performed using an SNR level of 50 and for two b-factors of 350 and 1000.0 s mm–2. The b-factor of 350 s mm–2 is optimal for the high-diffusion regions, as suggested by Xing et al. (34), since b * μ(D) ∼ 1.1. The b-factor of 1000 s mm–2 is close to the optimum for gray and white matter regions. To reduce the clutter, only the Icosa6 and Icosa21 are shown in Fig. 6. For clarity, Fig. 6d was zoomed and the results are shown in Fig. 7. Note that the estimation accuracy, as indicated by the peak height and width, is dependent on the SNR level, anisotropy, and b-factor for the different regions (17, 18). Note also that the DT and DW approaches yielded identical FA and μ(D) values for the Icosa6, consistent with our experimental results. The FA results for the Icosa21 scheme indicate that only CSF and edema regions are overestimated for b = 1000 s mm–2, mainly due to the suboptimal selection of the b-factor (b * μ(D) > 1.1) and the sensitivity of low-anisotropy regions to noise. Notice that 1) measurement noise creates an artifactual anisotropy for both these tissues with isotropic diffusion, which can be reduced by increasing the SNR and properly selecting the b-factor; and 2) due to the increase in SNR, μ(D) estimation is more immune to noise compared to FA. Note also that reducing the b-factor would optimize the isotropic CSF and edema tensor estimation at the expense of biasing the planar and axial region anisotropy estimates (17, 18, 34). Therefore, these simulations are completely consistent with our experimental observations.

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Figure 6. An illustration of the Monte Carlo simulation histogram plot results of FA and μ(D) as computed using the DT and DW methods. The numerical phantom shown in Fig. 2 is used with SNR = 50, and for the Icosa6 and Icosa21 encoding schemes. a: μ(D), b-factor = 350 s mm–2. b: FA, b-factor = 350 s mm–2. c: μ(D), b-factor = 1000 s mm–2. d: FA, b-factor = 1000 s mm–2.

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Figure 7. A zoomed version of the histogram plot in Fig. 6d. Note that there is no significant difference between FA(DT) and FA(DW) for the axial (a) and planar (b) anisotropic regions for both Icosa6 and Icosa21. The spherical tensor region (c) shows more sensitivity to the SNR level. The DW-based FA anisotropy measure appears to be less sensitive to noise, as indicated by the double peaks using Icosa21.

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DISCUSSION

  1. Top of page
  2. Abstract
  3. THEORY
  4. METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

The main purpose of these studies was to demonstrate theoretically, and verify experimentally and through simulations, that it is possible to derive μ(D), FA, and RA from the DW data without tensor decoding and diagonalization. This analysis, based on the single tensor model (1, 9, 10), shows that the relationships between the DW and DTI-measured schemes are strictly valid only for the icosahedral encoding scheme, which is rotationally invariant. The simulations also indicate that the results are valid for uniformly distributed encoding schemes at high SNR. The difference between the DT and DW anisotropy results for uniformly distributed icosahedral encoding schemes is attributed to the slightly different sensitivity of the two approaches to noise. For uniformly distributed encoding schemes with Ne > 6, the tensor estimation using the tensor model involves a least-squares optimization of the decoding matrix elements. In the DW-based approach, all measurements are treated on an equal footing. DT encoding schemes, such as S9, S12, and perturbed Icosa6, are not rotationally invariant and are likely to cause a large bias, as indicated by the results summarized in Fig. 3. This was observed in all four subjects.

An important result of these studies is that the value of FA derived from the DW data is related to the mean and SD of the DW measurements. These parameters can be rapidly computed and will eliminate the need for offline and computationally intensive calculations. Thus it is possible to display the results on the scanner to provide rapid feedback to the clinician, which can be of great help in the management of acute patients. Since the eigenvectors are not calculated, this approach is not applicable for vector-based tractography (3).

It should be mentioned that a number of published studies have demonstrated that it is possible to estimate the diffusion anisotropy from the DW data without the need to completely estimate, diagonalize, and analyze the tensor. For example, Ulug and van Zijl (35) introduced a host of rotationally invariant anisotropy measures that can be defined from the tensor invariants but still require full tensor estimation. Hasan et al. (30) showed that the eigenvectors, eigenvalues, and any eigenvalue-dependent anisotropy measure can be obtained directly from the DT elements and invariants. In that approach, the eigenvalues are obtained from RA (an aspherism index that is related to Aσ (35)), μ(D), and a surface-to-volume ratio measure (35). These approaches require tensor decoding with offline processing. Arfanakis et al. (12) showed that useful information can be obtained offline from higher moments of the DW data via spatially independent component analysis (ICA). Similarly, Frank (13, 20) showed that diffusion anisotropy maps can be computed using high-angular-resolution diffusion data without the need to resort to tensor analysis. Unlike the analysis presented in the current study, these methods all involve intensive offline processing. More recent studies (36, 37) (which were published while this manuscript was under review) also reported that the icosahedral encoding scheme can provide anisotropy estimates using the DW data. The current work presents a comprehensive theoretical and experimental analysis, and a detailed comparison between the DW- and tensor-based approaches.

The difference maps in Fig. 3 (third row) show a number of interesting features. For instance, the absolute μ(D) difference maps show nonzero intensity only for the perturbed icosahedron scheme for all of the b and SNR values. The theoretical analysis presented herein was developed using the principal icosahedron with Ne = 6, and was also validated using the Icosa15 and Icosa21 sets (Eq. [ 16]), which are both rotationally invariant and balanced. The perturbed icosahedron scheme differs from the other schemes in that it is both rotationally variant and unbalanced. However, this behavior is not seen for the S9 and S12 schemes, which are rotationally variant and balanced. Since this behavior was observed in all four subjects, the bias introduced in the computation of the μ(D) maps using the perturbed Icosa6 appears mainly to be the result of the unbalanced nature of the scheme. Another interesting observation is that the FA difference maps (Fig. 3, row 6) show nonzero intensity for all of the encoding schemes, except for the icosahedron encoding scheme with Ne = 6. This is not surprising, since the theoretical analysis is strictly valid only for the icosahedron encoding scheme with Ne = 6. What is surprising, and perhaps somewhat counterintuitive, is that the difference maps have nonzero intensity for gray matter and cerebrospinal fluid (CSF), but not for white matter. The Monte Carlo simulations presented here indicate that this behavior could be due to a suboptimal selection of the b-factor, which resulted in a poor estimation of the tensors with high diffusivity. In general, the error in the estimated FA depends on many variables, such as the SNR level, b-factor, encoding scheme, and DT attributes (e.g., orientation and anisotropy) (17, 18, 24–26, 34). In this work, a two-b-factor diffusion sensitization approach was adopted. A b-factor of 1000 s mm–2 may be optimal for white and gray matter, but in high-isotropic-diffusion regions this value would have not been optimal (34). This would result in an inaccurate estimation of the tensor attributes. The Monte Carlo simulations (Figs. 6 and 7) indicate that at high SNR, the value of FA can be computed directly from the DW data as long as the encoding scheme is uniformly distributed and icosahedral. A choice of low SNR levels and a high b-factor will result in poor estimation of the tensor attributes in regions of low anisotropy, such as CSF and gray matter (17, 18, 34). The use of fluid attenuated inversion recovery (FLAIR), thinner slice thickness, and cardiac pulse gating may ameliorate the effects of pulsatile motion and partial volume averaging in CSF (38).

The encoding schemes that were investigated in the current study can be broadly divided into four categories: 1) principal icosahedral with Ne = 6, which is rotationally invariant and balanced; 2) icosahedral with Ne = 15 and 21, which are rotationally invariant and balanced; 3) perturbed icosahedral with Ne = 6, which is rotationally variant and unbalanced; and 4) heuristic with Ne = 9 and 12, which are rotationally variant but balanced. As shown above, the theory was derived for icosahedral with Ne = 6, and validated using both Monte-Carlo simulations and experimental data. The theory is also valid for the icosahedral encoding schemes with Ne = 15 and 21. The anisotropy (but not μ(D)) does show a bias between the DW and DT methods; however, this bias is not an inherent property of these encoding schemes. In practice, the bias arises because Ne > 6 represents an overdetermined system of equations that requires a least-squares minimization approach. Thus, the DT-based approach tends to be more immune to the measurement noise compared to the DW approach. This bias depends on the SNR as well as the value of the anisotropy. This is consistent with both Monte-Carlo simulations and experimental data, as shown in Figs. 3–7. Therefore, this bias may limit the application of DW-based approaches for Ne > 6, low SNR, and low anisotropy. Investigations of the quantitative relationship between the bias and different variables (SNR, anisotropy values, encoding schemes, etc.) are currently in progress.

CONCLUSIONS

  1. Top of page
  2. Abstract
  3. THEORY
  4. METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

In this work, a novel approach to compute the rotationally invariant μ(D), RA, and FA anisotropy measures directly from DW data, without resorting to the tensor model, was introduced. The theoretical analysis was validated with both Monte Carlo simulations and normal human brain data acquired at different b-values, NEX, and SNR levels using a novel, multifaceted, tensor encoding scheme. The main requirements and limitations of this approach were discussed. A direct relationship between RA and FA was derived. This relationship, in combination with error propagation, can be used to consolidate a number of published reports that use either RA or FA. The implementation of the procedures presented in this work, combined with efficient eddy-current distortion-reduction strategies (such as the dual or twice refocused spin-echo approach (39, 40), would allow online computation of μ(D), RA, and FA using scanner software. This will also extend the utility of anisotropy maps to routine clinical applications for acute diseases. This work also demonstrated the importance of icosahedral-based sets and the role played by suboptimal rotationally variant and unbalanced sets.

Acknowledgements

  1. Top of page
  2. Abstract
  3. THEORY
  4. METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES

Discussions with Dr. Sajja Rao, and the editorial assistance of Marci Harris and Cheramy Goff were greatly appreciated.

REFERENCES

  1. Top of page
  2. Abstract
  3. THEORY
  4. METHODS
  5. RESULTS
  6. DISCUSSION
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES