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Estimation of Bounded and Unbounded Trajectories in Diffusion MRI.

Ning L, Westin CF, Rathi Y - Front Neurosci (2016)

Bottom Line: To this end, we propose to use the multivariate Ornstein-Uhlenbeck (OU) process to model the matrix-valued exponential autocorrelation function of three-dimensional diffusion processes with bounded trajectories.We present detailed analysis on the relation between the model parameters and the time-dependent apparent axon radius and provide a general model for dMRI signals from the frequency domain perspective.For our experimental setup, we model the diffusion signal as a mixture of two compartments that correspond to diffusing spins with bounded and unbounded trajectories, and analyze the corpus-callosum in an ex-vivo data set of a monkey brain.

View Article: PubMed Central - PubMed

Affiliation: Harvard Medical School, Brigham and Women's Hospital Boston, MA, USA.

ABSTRACT
Disentangling the tissue microstructural information from the diffusion magnetic resonance imaging (dMRI) measurements is quite important for extracting brain tissue specific measures. The autocorrelation function of diffusing spins is key for understanding the relation between dMRI signals and the acquisition gradient sequences. In this paper, we demonstrate that the autocorrelation of diffusion in restricted or bounded spaces can be well approximated by exponential functions. To this end, we propose to use the multivariate Ornstein-Uhlenbeck (OU) process to model the matrix-valued exponential autocorrelation function of three-dimensional diffusion processes with bounded trajectories. We present detailed analysis on the relation between the model parameters and the time-dependent apparent axon radius and provide a general model for dMRI signals from the frequency domain perspective. For our experimental setup, we model the diffusion signal as a mixture of two compartments that correspond to diffusing spins with bounded and unbounded trajectories, and analyze the corpus-callosum in an ex-vivo data set of a monkey brain.

No MeSH data available.


(A) shows the mean-squared displacements of the three data sets with different (δ, Δ) with the values shown in Figure 2F: the solid curves are the estimated results using the proposed model and the dashed lines are the corresponding results obtained from the DTI model, respectively. (B) shows the corresponding time-dependent diffusion coefficients for the three data sets.
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Figure 3: (A) shows the mean-squared displacements of the three data sets with different (δ, Δ) with the values shown in Figure 2F: the solid curves are the estimated results using the proposed model and the dashed lines are the corresponding results obtained from the DTI model, respectively. (B) shows the corresponding time-dependent diffusion coefficients for the three data sets.

Mentions: According to Equation (20), the estimated mean-squared displacements of the water molecules are given by pRou(δ, Δ) + (1 − p)RD(δ, Δ). In order to compare this model with DTI, we also computed the mean-squared displacements using DTI for each data set. The solid and dashed plots in Figure 3A show the estimated mean-squared displacements along the radial direction of the fiber bundles obtained using the proposed model and DTI, where the blue, green and red curves denote the average values from four voxels in the genu, midbody, and splenium areas, respectively. The solid plots show similar features as the dashed lines. Figure 3B shows the corresponding diffusion coefficients estimated using the proposed model and DTI, respectively, with the diffusion coefficients from the proposed model given by (pRou(δ, Δ) + (1 − p)RD(δ, Δ))/(Δ − δ/3). The reasons for the differences between the results include the measurement noise and that the proposed model (Equation 20) is non-Gaussian, i.e., a non-exponential function of the b-value, while DTI corresponds to a Gaussian model. The Gaussian model is not able to correctly estimate the diffusion coefficients at high b-values as in the data sets used in this experiment.


Estimation of Bounded and Unbounded Trajectories in Diffusion MRI.

Ning L, Westin CF, Rathi Y - Front Neurosci (2016)

(A) shows the mean-squared displacements of the three data sets with different (δ, Δ) with the values shown in Figure 2F: the solid curves are the estimated results using the proposed model and the dashed lines are the corresponding results obtained from the DTI model, respectively. (B) shows the corresponding time-dependent diffusion coefficients for the three data sets.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4814562&req=5

Figure 3: (A) shows the mean-squared displacements of the three data sets with different (δ, Δ) with the values shown in Figure 2F: the solid curves are the estimated results using the proposed model and the dashed lines are the corresponding results obtained from the DTI model, respectively. (B) shows the corresponding time-dependent diffusion coefficients for the three data sets.
Mentions: According to Equation (20), the estimated mean-squared displacements of the water molecules are given by pRou(δ, Δ) + (1 − p)RD(δ, Δ). In order to compare this model with DTI, we also computed the mean-squared displacements using DTI for each data set. The solid and dashed plots in Figure 3A show the estimated mean-squared displacements along the radial direction of the fiber bundles obtained using the proposed model and DTI, where the blue, green and red curves denote the average values from four voxels in the genu, midbody, and splenium areas, respectively. The solid plots show similar features as the dashed lines. Figure 3B shows the corresponding diffusion coefficients estimated using the proposed model and DTI, respectively, with the diffusion coefficients from the proposed model given by (pRou(δ, Δ) + (1 − p)RD(δ, Δ))/(Δ − δ/3). The reasons for the differences between the results include the measurement noise and that the proposed model (Equation 20) is non-Gaussian, i.e., a non-exponential function of the b-value, while DTI corresponds to a Gaussian model. The Gaussian model is not able to correctly estimate the diffusion coefficients at high b-values as in the data sets used in this experiment.

Bottom Line: To this end, we propose to use the multivariate Ornstein-Uhlenbeck (OU) process to model the matrix-valued exponential autocorrelation function of three-dimensional diffusion processes with bounded trajectories.We present detailed analysis on the relation between the model parameters and the time-dependent apparent axon radius and provide a general model for dMRI signals from the frequency domain perspective.For our experimental setup, we model the diffusion signal as a mixture of two compartments that correspond to diffusing spins with bounded and unbounded trajectories, and analyze the corpus-callosum in an ex-vivo data set of a monkey brain.

View Article: PubMed Central - PubMed

Affiliation: Harvard Medical School, Brigham and Women's Hospital Boston, MA, USA.

ABSTRACT
Disentangling the tissue microstructural information from the diffusion magnetic resonance imaging (dMRI) measurements is quite important for extracting brain tissue specific measures. The autocorrelation function of diffusing spins is key for understanding the relation between dMRI signals and the acquisition gradient sequences. In this paper, we demonstrate that the autocorrelation of diffusion in restricted or bounded spaces can be well approximated by exponential functions. To this end, we propose to use the multivariate Ornstein-Uhlenbeck (OU) process to model the matrix-valued exponential autocorrelation function of three-dimensional diffusion processes with bounded trajectories. We present detailed analysis on the relation between the model parameters and the time-dependent apparent axon radius and provide a general model for dMRI signals from the frequency domain perspective. For our experimental setup, we model the diffusion signal as a mixture of two compartments that correspond to diffusing spins with bounded and unbounded trajectories, and analyze the corpus-callosum in an ex-vivo data set of a monkey brain.

No MeSH data available.