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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.


Related in: MedlinePlus

(A–C) shows the estimated radius of bounded trajectories, the fraction of the bounded compartment in the dMRI signal and the normalized mean-square error between the estimated signal and the measurements, respectively. (D–F) shows the measured (dot points) signal and the estimated signal (solid lines) in three representative voxels from the genu, mid-body and splenium areas, respectively, with the horizontal axis being absolute value of the inner product between the gradient direction and the axon orientation.
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Figure 2: (A–C) shows the estimated radius of bounded trajectories, the fraction of the bounded compartment in the dMRI signal and the normalized mean-square error between the estimated signal and the measurements, respectively. (D–F) shows the measured (dot points) signal and the estimated signal (solid lines) in three representative voxels from the genu, mid-body and splenium areas, respectively, with the horizontal axis being absolute value of the inner product between the gradient direction and the axon orientation.

Mentions: For the estimated model parameters in each voxel, let c⊥ denote the smallest eigenvalue of C. Then, we define as the average radius of the bounded trajectories of the diffusing molecules. Figure 2A shows the estimated average trajectory radius () in the corpus-callosum of the monkey brain with the background being the standard fractional anisotropy (FA) image. We can see that the radius in the mid-body of the corpus-callosum is larger than the genu and the splenium. We note that a similar pattern for axon diameters has also been observed from histology analysis in mouse (Barazany et al., 2009), monkey and human brains (Caminiti et al., 2013). We note that the axon radius reported from histology studies of the monkey corpus-callosum range between 0.5−2.5μm, whereas the estimated values of are in the range 0.7−5μm. We conjecture that the larger value for the radius of spin trajectories compared with the axon size may be due to bounded diffusion processes with larger radii in densely packed extra axonal space. The estimated also reveals a pattern similar to the known distribution of axon sizes as larger axons will create more space for water molecules to diffuse, leading to trajectories with larger radii.


Estimation of Bounded and Unbounded Trajectories in Diffusion MRI.

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

(A–C) shows the estimated radius of bounded trajectories, the fraction of the bounded compartment in the dMRI signal and the normalized mean-square error between the estimated signal and the measurements, respectively. (D–F) shows the measured (dot points) signal and the estimated signal (solid lines) in three representative voxels from the genu, mid-body and splenium areas, respectively, with the horizontal axis being absolute value of the inner product between the gradient direction and the axon orientation.
© Copyright Policy
Related In: Results  -  Collection

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

Figure 2: (A–C) shows the estimated radius of bounded trajectories, the fraction of the bounded compartment in the dMRI signal and the normalized mean-square error between the estimated signal and the measurements, respectively. (D–F) shows the measured (dot points) signal and the estimated signal (solid lines) in three representative voxels from the genu, mid-body and splenium areas, respectively, with the horizontal axis being absolute value of the inner product between the gradient direction and the axon orientation.
Mentions: For the estimated model parameters in each voxel, let c⊥ denote the smallest eigenvalue of C. Then, we define as the average radius of the bounded trajectories of the diffusing molecules. Figure 2A shows the estimated average trajectory radius () in the corpus-callosum of the monkey brain with the background being the standard fractional anisotropy (FA) image. We can see that the radius in the mid-body of the corpus-callosum is larger than the genu and the splenium. We note that a similar pattern for axon diameters has also been observed from histology analysis in mouse (Barazany et al., 2009), monkey and human brains (Caminiti et al., 2013). We note that the axon radius reported from histology studies of the monkey corpus-callosum range between 0.5−2.5μm, whereas the estimated values of are in the range 0.7−5μm. We conjecture that the larger value for the radius of spin trajectories compared with the axon size may be due to bounded diffusion processes with larger radii in densely packed extra axonal space. The estimated also reveals a pattern similar to the known distribution of axon sizes as larger axons will create more space for water molecules to diffuse, leading to trajectories with larger radii.

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.


Related in: MedlinePlus