DWI acquisition schemes and Diffusion Tensor estimation A simulation based study Santiago Aja-Fernández, Antonio Tristán-Vega, Pablo Casaseca-de-la-Higuera Laboratory of Image Processing L A B O R A T O R I O D E P R O C E S A D O D E I M A G E N Universidad de Valladolid
Contents 1 Introduction 2 An introduction to diffusion MRI 3 Diffusion Tensor Estimation 4 Hypotheses about the acquistion modalities 5 Experiments and results 6 Conclusions
Contents 1 Introduction 2 An introduction to diffusion MRI 3 Diffusion Tensor Estimation 4 Hypotheses about the acquistion modalities 5 Experiments and results 6 Conclusions Laboratory of Image Processing (LPI) DWI adquisition and DTI 3 / 22
Introduction Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) well-known established imaging modality used to measure water diffusivity in tissues. Based on Diffusion tensor (DT): Least Squares (LS) de facto standard to estimate DT. [Salvador05] and [Tristan09] theoretical framework to estimate DT from single- and multiple-coil systems using Weighted LS. Simulation-based study of the effect of the acquisition schemes over the estimated DT. Four cases: single coil acquisition, multiple coil acquisition (no subsampling) and subsampled multiple coil acquisition with pmri reconstruction (SENSE and GRAPPA). Laboratory of Image Processing (LPI) DWI adquisition and DTI 4 / 22
Introduction Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) well-known established imaging modality used to measure water diffusivity in tissues. Based on Diffusion tensor (DT): Least Squares (LS) de facto standard to estimate DT. [Salvador05] and [Tristan09] theoretical framework to estimate DT from single- and multiple-coil systems using Weighted LS. Simulation-based study of the effect of the acquisition schemes over the estimated DT. Four cases: single coil acquisition, multiple coil acquisition (no subsampling) and subsampled multiple coil acquisition with pmri reconstruction (SENSE and GRAPPA). Laboratory of Image Processing (LPI) DWI adquisition and DTI 4 / 22
Introduction Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) well-known established imaging modality used to measure water diffusivity in tissues. Based on Diffusion tensor (DT): Least Squares (LS) de facto standard to estimate DT. [Salvador05] and [Tristan09] theoretical framework to estimate DT from single- and multiple-coil systems using Weighted LS. Simulation-based study of the effect of the acquisition schemes over the estimated DT. Four cases: single coil acquisition, multiple coil acquisition (no subsampling) and subsampled multiple coil acquisition with pmri reconstruction (SENSE and GRAPPA). Laboratory of Image Processing (LPI) DWI adquisition and DTI 4 / 22
Introduction Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) well-known established imaging modality used to measure water diffusivity in tissues. Based on Diffusion tensor (DT): Least Squares (LS) de facto standard to estimate DT. [Salvador05] and [Tristan09] theoretical framework to estimate DT from single- and multiple-coil systems using Weighted LS. Simulation-based study of the effect of the acquisition schemes over the estimated DT. Four cases: single coil acquisition, multiple coil acquisition (no subsampling) and subsampled multiple coil acquisition with pmri reconstruction (SENSE and GRAPPA). [Salvador05] R. Salvador et al., Formal characterization and extension of the linearized diffusion tensor model, Human brain mapping, vol. 24, 2005. [Tristan09] A. Tristán-Vega, C.-F. Westin, and S. Aja-Fernández, Bias of least squares approaches for diffusion tensor estimation from array coils in DT MRI, in MICCAI 2009, 2009. Laboratory of Image Processing (LPI) DWI adquisition and DTI 4 / 22
Introduction Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) well-known established imaging modality used to measure water diffusivity in tissues. Based on Diffusion tensor (DT): Least Squares (LS) de facto standard to estimate DT. [Salvador05] and [Tristan09] theoretical framework to estimate DT from single- and multiple-coil systems using Weighted LS. Simulation-based study of the effect of the acquisition schemes over the estimated DT. Four cases: single coil acquisition, multiple coil acquisition (no subsampling) and subsampled multiple coil acquisition with pmri reconstruction (SENSE and GRAPPA). [Salvador05] R. Salvador et al., Formal characterization and extension of the linearized diffusion tensor model, Human brain mapping, vol. 24, 2005. [Tristan09] A. Tristán-Vega, C.-F. Westin, and S. Aja-Fernández, Bias of least squares approaches for diffusion tensor estimation from array coils in DT MRI, in MICCAI 2009, 2009. Laboratory of Image Processing (LPI) DWI adquisition and DTI 4 / 22
Contents 1 Introduction 2 An introduction to diffusion MRI 3 Diffusion Tensor Estimation 4 Hypotheses about the acquistion modalities 5 Experiments and results 6 Conclusions Laboratory of Image Processing (LPI) DWI adquisition and DTI 5 / 22
Diffusion Tensor MRI MRI versus dmri Main idea behind Diffusion Tensor MRI MRI: distinguish tissues dmri: track nerve fibers The white matter is strongly anisotropic This anisotropy can be indirectly characterized through the diffusion of water molecules A diffusion tensor is assigned to each 3D pixel : Rank-2 tensor 3 3 symmetric matrix Eigenvectors: principal diffusion directions Eigenvalues: amount of diffusion along them Laboratory of Image Processing (LPI) DWI adquisition and DTI 6 / 22
Diffusion Tensor MRI MRI versus dmri Main idea behind Diffusion Tensor MRI MRI: distinguish tissues dmri: track nerve fibers The white matter is strongly anisotropic This anisotropy can be indirectly characterized through the diffusion of water molecules A diffusion tensor is assigned to each 3D pixel : Rank-2 tensor 3 3 symmetric matrix Eigenvectors: principal diffusion directions Eigenvalues: amount of diffusion along them Laboratory of Image Processing (LPI) DWI adquisition and DTI 6 / 22
Diffusion Tensor MRI MRI versus dmri Main idea behind Diffusion Tensor MRI MRI: distinguish tissues dmri: track nerve fibers The white matter is strongly anisotropic This anisotropy can be indirectly characterized through the diffusion of water molecules A diffusion tensor is assigned to each 3D pixel : Rank-2 tensor 3 3 symmetric matrix Eigenvectors: principal diffusion directions Eigenvalues: amount of diffusion along them Laboratory of Image Processing (LPI) DWI adquisition and DTI 6 / 22
Diffusion Tensor MRI MRI versus dmri Main idea behind Diffusion Tensor MRI MRI: distinguish tissues dmri: track nerve fibers The white matter is strongly anisotropic This anisotropy can be indirectly characterized through the diffusion of water molecules A diffusion tensor is assigned to each 3D pixel : Rank-2 tensor 3 3 symmetric matrix Eigenvectors: principal diffusion directions Eigenvalues: amount of diffusion along them Laboratory of Image Processing (LPI) DWI adquisition and DTI 6 / 22
Interpretation of DT-MRI data Scalar measures and beyond Scalar magnitudes derived from the eigenvalues Fractional anisotropy: std(λi ) Mean diffusivity: λi Vector measures derived from the eigenvalues and eigenvectors Color coding: (ex, ey, ez ) (R, G, B) Glyphs Full characterization of diffusion: fiber tracking Laboratory of Image Processing (LPI) DWI adquisition and DTI 7 / 22
Interpretation of DT-MRI data Scalar measures and beyond Scalar magnitudes derived from the eigenvalues Fractional anisotropy: std(λi ) Mean diffusivity: λi Vector measures derived from the eigenvalues and eigenvectors Color coding: (ex, ey, ez ) (R, G, B) Glyphs Full characterization of diffusion: fiber tracking Laboratory of Image Processing (LPI) DWI adquisition and DTI 7 / 22
Interpretation of DT-MRI data Scalar measures and beyond Scalar magnitudes derived from the eigenvalues Fractional anisotropy: std(λi ) Mean diffusivity: λi Vector measures derived from the eigenvalues and eigenvectors Color coding: (ex, ey, ez ) (R, G, B) Glyphs Full characterization of diffusion: fiber tracking Laboratory of Image Processing (LPI) DWI adquisition and DTI 7 / 22
Interpretation of DT-MRI data Scalar measures and beyond Scalar magnitudes derived from the eigenvalues Fractional anisotropy: std(λi ) Mean diffusivity: λi Vector measures derived from the eigenvalues and eigenvectors Color coding: (ex, ey, ez ) (R, G, B) Glyphs Full characterization of diffusion: fiber tracking Laboratory of Image Processing (LPI) DWI adquisition and DTI 7 / 22
Interpretation of DT-MRI data Scalar measures and beyond Scalar magnitudes derived from the eigenvalues Fractional anisotropy: std(λi ) Mean diffusivity: λi Vector measures derived from the eigenvalues and eigenvectors Color coding: (ex, ey, ez ) (R, G, B) Glyphs Full characterization of diffusion: fiber tracking Laboratory of Image Processing (LPI) DWI adquisition and DTI 7 / 22
Contents 1 Introduction 2 An introduction to diffusion MRI 3 Diffusion Tensor Estimation 4 Hypotheses about the acquistion modalities 5 Experiments and results 6 Conclusions Laboratory of Image Processing (LPI) DWI adquisition and DTI 8 / 22
The diffusion tensor (DT) Most dmri applications based on single tensor approach. Known weakness of single tensor: fiber crossing or kissing. Least Squares (LS) techniques de facto standard: Weighted Least Squares (WLS). The big question: To use more gradient directions or more repetitions (NEX)? In [Tristan09]: variance in the estimation reduced by increasing the number of gradient directions. Bias reduced by increasing NEX. Precission in estimation depends on underlaying Statistical model Laboratory of Image Processing (LPI) DWI adquisition and DTI 9 / 22
The diffusion tensor (DT) Most dmri applications based on single tensor approach. Known weakness of single tensor: fiber crossing or kissing. Least Squares (LS) techniques de facto standard: Weighted Least Squares (WLS). The big question: To use more gradient directions or more repetitions (NEX)? In [Tristan09]: variance in the estimation reduced by increasing the number of gradient directions. Bias reduced by increasing NEX. Precission in estimation depends on underlaying Statistical model Laboratory of Image Processing (LPI) DWI adquisition and DTI 9 / 22
The diffusion tensor (DT) Most dmri applications based on single tensor approach. Known weakness of single tensor: fiber crossing or kissing. Least Squares (LS) techniques de facto standard: Weighted Least Squares (WLS). The big question: To use more gradient directions or more repetitions (NEX)? In [Tristan09]: variance in the estimation reduced by increasing the number of gradient directions. Bias reduced by increasing NEX. Precission in estimation depends on underlaying Statistical model Laboratory of Image Processing (LPI) DWI adquisition and DTI 9 / 22
The diffusion tensor (DT) Most dmri applications based on single tensor approach. Known weakness of single tensor: fiber crossing or kissing. Least Squares (LS) techniques de facto standard: Weighted Least Squares (WLS). The big question: To use more gradient directions or more repetitions (NEX)? In [Tristan09]: variance in the estimation reduced by increasing the number of gradient directions. Bias reduced by increasing NEX. Precission in estimation depends on underlaying Statistical model Laboratory of Image Processing (LPI) DWI adquisition and DTI 9 / 22
The diffusion tensor (DT) Most dmri applications based on single tensor approach. Known weakness of single tensor: fiber crossing or kissing. Least Squares (LS) techniques de facto standard: Weighted Least Squares (WLS). The big question: To use more gradient directions or more repetitions (NEX)? In [Tristan09]: variance in the estimation reduced by increasing the number of gradient directions. Bias reduced by increasing NEX. Precission in estimation depends on underlaying Statistical model Laboratory of Image Processing (LPI) DWI adquisition and DTI 9 / 22
The diffusion tensor (DT) Most dmri applications based on single tensor approach. Known weakness of single tensor: fiber crossing or kissing. Least Squares (LS) techniques de facto standard: Weighted Least Squares (WLS). The big question: To use more gradient directions or more repetitions (NEX)? In [Tristan09]: variance in the estimation reduced by increasing the number of gradient directions. Bias reduced by increasing NEX. Precission in estimation depends on underlaying Statistical model Laboratory of Image Processing (LPI) DWI adquisition and DTI 9 / 22
Signal and noise statistical models in MR Number of Adquisition Statistical Model Stat. model of the coils background 1 coil Single coil Rician Rayleigh (Stationary) Multiple coils No subsampling+ Non-central Chi Central Chi SoS (Non-Stationary) Multiple coils pmri+ Rician Rayleigh SENSE (Non-Stationary) Multiple coils pmri+ GRAPPA+ SoS Non-central Chi Central Chi (Non-Stationary, effective parameters) Laboratory of Image Processing (LPI) DWI adquisition and DTI 10 / 22
Tensor fitting based on Weighted Least Squares Estimation error for a simplified scenario [Tristan09] The error (MSE) for multiple-coil is defined as [ ( K1 1 MSE N SNR 2 1 )] (3L 4) 4 SNR }{{} Var(estimation) [ + ] 1 3(L 1)2 4 SNR }{{} bias 2 (estimation) Bias Real Value Standard deviation Measurement Real Value Measurement Measurement Real Value Laboratory of Image Processing (LPI) DWI adquisition and DTI 11 / 22
Contents 1 Introduction 2 An introduction to diffusion MRI 3 Diffusion Tensor Estimation 4 Hypotheses about the acquistion modalities 5 Experiments and results 6 Conclusions Laboratory of Image Processing (LPI) DWI adquisition and DTI 12 / 22
Hypotheses about the acquistion modalities 1 The contribution of the bias will be much smaller for the Rician than for the nc-χ case. Accordingly, the bias on the estimation will be smaller in one-coil systems and in pmri when reconstructed with SENSE. 2 The error due to the variance will be reduced by increasing the number of gradients (N). However, this is not the case for the bias, which cannot be reduced from taking more gradients. The only way to do so is by increasing the SNR. This can be done by properly filtering the data or by increasing the NEX (number of repetitions in the scanner). 3 pmri reconstructed images are known to decrease the SNR according to the so-called g-factor. So, the variance and bias of the error are expected to grow in these cases when compared to the fully-sampled multiple coil case. 4 Finally, equation does not take into account one additional source of error: in subsampled pmri the missing data must be estimated, i.e. the spatial aliasing must be corrected. The error in this estimation will also propagate to the DT estimation. Laboratory of Image Processing (LPI) DWI adquisition and DTI 13 / 22
Hypotheses about the acquistion modalities 1 The contribution of the bias will be much smaller for the Rician than for the nc-χ case. Accordingly, the bias on the estimation will be smaller in one-coil systems and in pmri when reconstructed with SENSE. 2 The error due to the variance will be reduced by increasing the number of gradients (N). However, this is not the case for the bias, which cannot be reduced from taking more gradients. The only way to do so is by increasing the SNR. This can be done by properly filtering the data or by increasing the NEX (number of repetitions in the scanner). 3 pmri reconstructed images are known to decrease the SNR according to the so-called g-factor. So, the variance and bias of the error are expected to grow in these cases when compared to the fully-sampled multiple coil case. 4 Finally, equation does not take into account one additional source of error: in subsampled pmri the missing data must be estimated, i.e. the spatial aliasing must be corrected. The error in this estimation will also propagate to the DT estimation. Laboratory of Image Processing (LPI) DWI adquisition and DTI 13 / 22
Hypotheses about the acquistion modalities 1 The contribution of the bias will be much smaller for the Rician than for the nc-χ case. Accordingly, the bias on the estimation will be smaller in one-coil systems and in pmri when reconstructed with SENSE. 2 The error due to the variance will be reduced by increasing the number of gradients (N). However, this is not the case for the bias, which cannot be reduced from taking more gradients. The only way to do so is by increasing the SNR. This can be done by properly filtering the data or by increasing the NEX (number of repetitions in the scanner). 3 pmri reconstructed images are known to decrease the SNR according to the so-called g-factor. So, the variance and bias of the error are expected to grow in these cases when compared to the fully-sampled multiple coil case. 4 Finally, equation does not take into account one additional source of error: in subsampled pmri the missing data must be estimated, i.e. the spatial aliasing must be corrected. The error in this estimation will also propagate to the DT estimation. Laboratory of Image Processing (LPI) DWI adquisition and DTI 13 / 22
Hypotheses about the acquistion modalities 1 The contribution of the bias will be much smaller for the Rician than for the nc-χ case. Accordingly, the bias on the estimation will be smaller in one-coil systems and in pmri when reconstructed with SENSE. 2 The error due to the variance will be reduced by increasing the number of gradients (N). However, this is not the case for the bias, which cannot be reduced from taking more gradients. The only way to do so is by increasing the SNR. This can be done by properly filtering the data or by increasing the NEX (number of repetitions in the scanner). 3 pmri reconstructed images are known to decrease the SNR according to the so-called g-factor. So, the variance and bias of the error are expected to grow in these cases when compared to the fully-sampled multiple coil case. 4 Finally, equation does not take into account one additional source of error: in subsampled pmri the missing data must be estimated, i.e. the spatial aliasing must be corrected. The error in this estimation will also propagate to the DT estimation. Laboratory of Image Processing (LPI) DWI adquisition and DTI 13 / 22
Contents 1 Introduction 2 An introduction to diffusion MRI 3 Diffusion Tensor Estimation 4 Hypotheses about the acquistion modalities 5 Experiments and results 6 Conclusions Laboratory of Image Processing (LPI) DWI adquisition and DTI 14 / 22
Synthetic experiments (a) Original (b) Rician (c) pmri-sos (d) SENSE (e) GRAPPA Synthetic 2D Tensor Field 2D used for the first experiment. Original size 128 128. Original and noisy cases, 8 coils, 5 gradients and σ n = 35. Laboratory of Image Processing (LPI) DWI adquisition and DTI 15 / 22
Synthetic experiments x 10 3 3 Single coil 8000 7000 x 10 3 3 pmri SoS 10000 9000 x 10 3 3 SENSE 1000 x 10 3 3 GRAPPA 1200 1000 λ 1 2.5 2 1.5 1 0.5 6000 5000 4000 3000 2000 1000 λ 1 2.5 2 1.5 1 0.5 8000 7000 6000 5000 4000 3000 2000 1000 λ 1 2.5 2 1.5 1 0.5 800 600 400 200 λ 1 2.5 2 1.5 1 0.5 800 600 400 200 5 0 5 10 15 λ 2 x 10 4 0 5 0 5 10 15 λ 2 x 10 4 0 5 0 5 10 15 λ 2 x 10 4 0 5 0 5 10 15 λ 2 x 10 4 0 3 x 10 3 Single coil x 10 4 16 3 x 10 3 pmri SoS x 10 4 16 3 x 10 3 SENSE 12000 3 x 10 3 GRAPPA 5000 2.5 14 12 2.5 14 12 2.5 10000 2.5 4500 4000 2 10 2 10 2 8000 2 3500 3000 λ 1 1.5 1 8 6 4 2 λ 1 1.5 1 8 6 4 2 λ 1 1.5 1 6000 4000 2000 λ 1 1.5 1 2500 2000 1500 1000 500 0.5 0.5 0.5 0 0 0.5 0 5 0 5 10 15 20 5 0 5 10 15 20 5 0 5 10 15 20 5 0 5 10 15 20 λ 2 λ 2 λ 2 λ 2 x 10 4 x 10 4 x 10 4 x 10 4 (a) Rician (b) pmri-sos (c) SENSE (d) GRAPPA 0 Two dimensional histograms of the distribution of the estimated eigenvalues: λ 1 vs. λ 2. Top row: σ n = 35 and 5 gradient directions. Low row: σ n = 10 and 15 gradient directions. In green the original eigenvalues. Laboratory of Image Processing (LPI) DWI adquisition and DTI 16 / 22
Realistic DWI phantom A realistic DWI phantom is used, [Tristan09b]. A 256 256 81 volume, spatial resolution of 1mm 1mm 1.7mm, 15 gradient directions and 1 baseline. [Tristan09b] A. Tristán-Vega and S. Aja-Fernández, Design and construction of a realistic DWI phantom for filtering performance assessment, in MICCAI 2009, 2009. Laboratory of Image Processing (LPI) DWI adquisition and DTI 17 / 22
Realistic DWI phantom rmse 2.5 2 1.5 1 0.5 pmri SoS GRAPPA Rician SENSE 0 5 10 15 20 25 30 35 σ n rmse 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 5 10 15 20 Number of Gradients pmri SoS GRAPPA Rician SENSE rmse of the tensor estimation for (Left) different σ n values (and 5 gradients); (Right) different number of gradients (and σ n = 10) rmse(x) = ( λ 1 (x) λ 1 (x)) 2 + ( λ 2 (x) λ 2 (x)) 2 + ( λ 3 (x) λ 3 (x)) 2 λ 1 (x) Laboratory of Image Processing (LPI) DWI adquisition and DTI 18 / 22
Realistic DWI phantom Fractional Anisotropy. From left to right: Original non-noisy data; Rician case; pmri-sos case; SENSE case; GRAPPA case. Top row σ n = 10 (average SNR in gray matter in the gradient images 40). Low row: σ n = 35 (average SNR in gray matter in the gradient images 11.4). Laboratory of Image Processing (LPI) DWI adquisition and DTI 19 / 22
Contents 1 Introduction 2 An introduction to diffusion MRI 3 Diffusion Tensor Estimation 4 Hypotheses about the acquistion modalities 5 Experiments and results 6 Conclusions Laboratory of Image Processing (LPI) DWI adquisition and DTI 20 / 22
Conclusions The methods considered can be modeled or approximated by a nc-χ distribution (Rician being a particular case). Three variables must be taken into account: the number of coils (that mainly affects to the bias in the error), the number of gradients (that mainly affects to the variance of the estimation error) and the SNR that globally affects the estimation error. Methods using multiple coils show a greater bias. pmri: accelerate the acquisition process but worsening the SNR: greater variance and bias in DT estimation. Laboratory of Image Processing (LPI) DWI adquisition and DTI 21 / 22
Conclusions The methods considered can be modeled or approximated by a nc-χ distribution (Rician being a particular case). Three variables must be taken into account: the number of coils (that mainly affects to the bias in the error), the number of gradients (that mainly affects to the variance of the estimation error) and the SNR that globally affects the estimation error. Methods using multiple coils show a greater bias. pmri: accelerate the acquisition process but worsening the SNR: greater variance and bias in DT estimation. Laboratory of Image Processing (LPI) DWI adquisition and DTI 21 / 22
Conclusions The methods considered can be modeled or approximated by a nc-χ distribution (Rician being a particular case). Three variables must be taken into account: the number of coils (that mainly affects to the bias in the error), the number of gradients (that mainly affects to the variance of the estimation error) and the SNR that globally affects the estimation error. Methods using multiple coils show a greater bias. pmri: accelerate the acquisition process but worsening the SNR: greater variance and bias in DT estimation. Laboratory of Image Processing (LPI) DWI adquisition and DTI 21 / 22
Conclusions The methods considered can be modeled or approximated by a nc-χ distribution (Rician being a particular case). Three variables must be taken into account: the number of coils (that mainly affects to the bias in the error), the number of gradients (that mainly affects to the variance of the estimation error) and the SNR that globally affects the estimation error. Methods using multiple coils show a greater bias. pmri: accelerate the acquisition process but worsening the SNR: greater variance and bias in DT estimation. Laboratory of Image Processing (LPI) DWI adquisition and DTI 21 / 22
DWI acquisition schemes and Diffusion Tensor estimation A simulation based study Santiago Aja-Fernández, Antonio Tristán-Vega, Pablo Casaseca-de-la-Higuera Laboratory of Image Processing L A B O R A T O R I O D E P R O C E S A D O D E I M A G E N Universidad de Valladolid