Limits...
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.


The logarithm of the positional autocorrelation function of diffusing particles in cylinders with radii varying from 1 to 8 μm. The plots show that the autocorrelation functions decrease approximately as exponential functions of diffusion time.
© Copyright Policy
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4814562&req=5

Figure 1: The logarithm of the positional autocorrelation function of diffusing particles in cylinders with radii varying from 1 to 8 μm. The plots show that the autocorrelation functions decrease approximately as exponential functions of diffusion time.

Mentions: In the case for restricted and impermeable pores the expression for the diffusion signal has been derived from the solution of diffusion equation with suitable boundary conditions (Stejskal and Tanner, 1965; Murday and Cotts, 1968; Neuman, 1974; Vangelderen et al., 1994; Åslund and Topgaard, 2009). In particular, the position autocorrelation function along the perpendicular direction of the restricted wall is given by Stepišnik (1993):(5)cr⊥(t,s)=2∑m=1∞r2αm2(αm2+1-n)e-αm2d⊥r2/t-s/,where n denotes the number of restricted dimensions (plane: n = 1, cylinder n = 2, sphere: n = 3), and αm is the m-th root of Jn/2(α) − αJ1+n/2(α) = 0, with Jv being the v-th order Bessel function of the first kind, d⊥ denotes the diffusivity along the perpendicular direction of the restricted walls. The mean-squared displacement of COMs is given by Neuman (1974):4δ2∑m=1∞r2αm2(αm2+1-n)f(αm2d⊥r2,δ,Δ),where the function f(·, ·, ·) is defined as(6)f(a,δ,Δ)=(2e−aδ+2e−aΔ−e−a(Δ+δ)−e−a(Δ−δ)                  −2+2aδ)a−2.We note that the autocorrelation function in Equation (5) is an infinite series of weighted exponential functions. As m increases, the weighting coefficients get smaller and the exponential function decays quickly. As a result, for long-time scales, the correlation is dominated by the first term corresponding to α1. For example, the first three values of αm are approximately given by α1 = 1.84, α2 = 5.33 and α3 = 8.54. Assuming r = 1μm, D = 1μm2/ms, t = 0.1 ms and n = 2, the first three terms of Equation (5) are given by 0.1759, 0.0013 and 5.63 × 10−6. Thus, the contribution of the terms for m ≥ 2 is very minimal. To show that the correlation function is approximately an exponential function, we plot the logarithm of cr(t, 0) with r = 1, 2, …8μm in Figure 1 where we include the first 500 terms (m = 1, …, 500) in the sum in Equation (5). The linear curves imply that the correlation function of the diffusion process in restricted pores can very well be approximated by a single exponential function, as obtained from our analysis in Equation (13). The validation of the exponential form of the autocorrelation function has also been discussed in Sheltraw and Kenkre (1996) and Burcaw et al. (2015).


Estimation of Bounded and Unbounded Trajectories in Diffusion MRI.

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

The logarithm of the positional autocorrelation function of diffusing particles in cylinders with radii varying from 1 to 8 μm. The plots show that the autocorrelation functions decrease approximately as exponential functions of diffusion time.
© Copyright Policy
Related In: Results  -  Collection

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

Figure 1: The logarithm of the positional autocorrelation function of diffusing particles in cylinders with radii varying from 1 to 8 μm. The plots show that the autocorrelation functions decrease approximately as exponential functions of diffusion time.
Mentions: In the case for restricted and impermeable pores the expression for the diffusion signal has been derived from the solution of diffusion equation with suitable boundary conditions (Stejskal and Tanner, 1965; Murday and Cotts, 1968; Neuman, 1974; Vangelderen et al., 1994; Åslund and Topgaard, 2009). In particular, the position autocorrelation function along the perpendicular direction of the restricted wall is given by Stepišnik (1993):(5)cr⊥(t,s)=2∑m=1∞r2αm2(αm2+1-n)e-αm2d⊥r2/t-s/,where n denotes the number of restricted dimensions (plane: n = 1, cylinder n = 2, sphere: n = 3), and αm is the m-th root of Jn/2(α) − αJ1+n/2(α) = 0, with Jv being the v-th order Bessel function of the first kind, d⊥ denotes the diffusivity along the perpendicular direction of the restricted walls. The mean-squared displacement of COMs is given by Neuman (1974):4δ2∑m=1∞r2αm2(αm2+1-n)f(αm2d⊥r2,δ,Δ),where the function f(·, ·, ·) is defined as(6)f(a,δ,Δ)=(2e−aδ+2e−aΔ−e−a(Δ+δ)−e−a(Δ−δ)                  −2+2aδ)a−2.We note that the autocorrelation function in Equation (5) is an infinite series of weighted exponential functions. As m increases, the weighting coefficients get smaller and the exponential function decays quickly. As a result, for long-time scales, the correlation is dominated by the first term corresponding to α1. For example, the first three values of αm are approximately given by α1 = 1.84, α2 = 5.33 and α3 = 8.54. Assuming r = 1μm, D = 1μm2/ms, t = 0.1 ms and n = 2, the first three terms of Equation (5) are given by 0.1759, 0.0013 and 5.63 × 10−6. Thus, the contribution of the terms for m ≥ 2 is very minimal. To show that the correlation function is approximately an exponential function, we plot the logarithm of cr(t, 0) with r = 1, 2, …8μm in Figure 1 where we include the first 500 terms (m = 1, …, 500) in the sum in Equation (5). The linear curves imply that the correlation function of the diffusion process in restricted pores can very well be approximated by a single exponential function, as obtained from our analysis in Equation (13). The validation of the exponential form of the autocorrelation function has also been discussed in Sheltraw and Kenkre (1996) and Burcaw et al. (2015).

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.