Pore size distribution estimation using the mixing time dependency of ...
Pore size distribution estimation using the mixing time dependency of a double diffusion encoding experiment: a proof of concept from Monte Carlo simulated data V. Methot and M. A. Koch Institute of Medical Engineering, University of Lübeck, Lübeck, Germany
Double diffusion encoding (DDE) MR [1] is sensitive to the dimension of microscopical pores [2]. The aquired signal (S) depends on the angle (ψ) and delay (mixing time: tm) between the two diffusion encodings [1,3] (Figure 1). We study the tm-dependency as a way to estimatimate spherical pore radii.
a
G1 Δ
δ
b
G2
tm δ
G1
Δ
G1
Δ
tm Δ
δ
G1
δ
δ
δ
Table 1: Simulation parameters
Monte Carlo simulated data from a random walk process were obtained using IDL (Exelis Visual Information Solutions, Boulder, CO, USA) [5] with the parameters given by Table 1. Inverse Laplace transform was done using Tikhonov regularized least squares with a non-negativity constraint using Scipy [5].
Time steps / ms
0.1
Number of paths
10 000
radii(a) / µm
5, 10, 15, 20, 25
/ ms
30
/ ms
1
G1, G2 / mT m-1
300
t m range / ms
0 – 200
G2
G2
G2
Results
Figure 1: Effective diffusion gradient waveforms for (a) parallel (ψ = 0) and (b) antiparallel (ψ = π) DDE.
From equation 1, the difference between parallel (Figure 1a) and antiparallel (Figure 1b) diffusion encodings can be reduce to: (eq. 2)
Parallel Antiparallel
0.4
S / S0
α : 1st root of 1st order spherical Bessel function's derivative E(a) : weighting function, can be obtained via simulations S0 : Non diffusion-weighted signal D0 : diffusion coefficient
0.3
ΔS 0.2 0.1 10
30 tm / ms
50
Spherical pores of fixed radius can be measured using an exponential fit with parameter ω = α2D0/a2 and variable tm. For a mixture of different radii, the inverse Laplace transform yields a volume fraction distribution. Corresponding Author: Vincent Méthot Institute of Medical Engineering, University of Lübeck Ratzeburger Allee 160, 23562 Lübeck, Germany [email protected]
350
a
5 µm radius 10 µm radius 15 µm radius 20 µm radius 25 µm radius
Monte Carlo simulated data
0.4 Signal reconstructed from the estimated pore size distribution
0.2 Bi exponential decay (expected signal) 0.05
0.8
b
1.2
0.10
0.15
D0 = 2e 3/ ¹m 2 s −1 α = 2.08
300
Expected radii
b
tm / s
0.20
Cartoon of the voxel
1.0 0.8
Estimated pore volume fraction
0.6 0.4 0.2
5 µm
0.0 100
20 µm
101
Spherical pore radius / µm
102
250 Theoretical relationship (according to [3]) 0.6
ω=
200 Exponential fit
ω / s-1
For spherical pores, the normalized signal can be approximated by [3]: (eq. 1)
Parallel - antiparallel signal difference Δ S / a.u.
Theory
0.6
1.4
Figure 2 shows the signal difference simulated for various pore radii. Figure 3 shows the estimated sphere radius distribution from a combination of ΔS obtained with a = 5 µm and a = 20 µm pores. 1.0
a
0.8
0.0 0.00
Volume fraction [a. u.]
δ
time
Methods
1.0
Signal difference Δ S
Introduction
Δ S ∝ e −ω t m
0.4
Figure 3: (a) Sum of simulated ΔS for 5 µm and 20 µm spherical pore radii as well as theoretical and fitted decays. (b) solution of the Laplace inversion for the simulated data.
α2 D0 a2
150
Discussion
Monte Carlo simulated data 100
The simulated data exhibit the expected exponential decay with t m. Furthermore, it is possible to use this mixing-time dependency to retrieve information about a distribution of spherical pore radii by inverse Laplace transform.
Fitted ω from Figure 2a 0.2 50
0.0
0 0
50
100
Mixing time tm / ms
150
200
0
5
10
15
20
25
30
Pore radius (a) / µm
Figure 2: (a) Simulated ΔS for various spherical pore size radii and corresponding exponential fit. (b) Fitted decay parameter ω (from (a)) compared to theoretical prediction as in [3] (see eq. 2).
Funding: German Research Foundation, grant KO3389/2-1
References: [1] P. P. Mitra. Phys. Rev. B 51:15074–15078 (1995). [4] M. A. Koch and J. Finsterbusch. Magn. Reson. Med. 62:247–254 (2009). [2] D. Benjamini and U. Nevo. J. Magn. Reson. 230:198–204 (2013). [5] S. van der Walt et al. Comput. Sci. Eng. 13:22-30 (2011). [3] E. Özarslan and P. J. Basser. J. Chem. Phys. 128:154511 (2008).
Double diffusion encoding (DDE) MR [1] is sensitive to the dimension of microscopical pores [2]. The aquired signal (S) depends on the angle (ψ) and delay (mixing time: tm) between the two diffusion encodings [1,3] (Figure 1). We study the tm-dependency as a way to estimatimate spherical pore radii.
a
G1 Δ
δ
b
G2
tm δ
G1
Δ
G1
Δ
tm Δ
δ
G1
δ
δ
δ
Table 1: Simulation parameters
Monte Carlo simulated data from a random walk process were obtained using IDL (Exelis Visual Information Solutions, Boulder, CO, USA) [5] with the parameters given by Table 1. Inverse Laplace transform was done using Tikhonov regularized least squares with a non-negativity constraint using Scipy [5].
Time steps / ms
0.1
Number of paths
10 000
radii(a) / µm
5, 10, 15, 20, 25
/ ms
30
/ ms
1
G1, G2 / mT m-1
300
t m range / ms
0 – 200
G2
G2
G2
Results
Figure 1: Effective diffusion gradient waveforms for (a) parallel (ψ = 0) and (b) antiparallel (ψ = π) DDE.
From equation 1, the difference between parallel (Figure 1a) and antiparallel (Figure 1b) diffusion encodings can be reduce to: (eq. 2)
Parallel Antiparallel
0.4
S / S0
α : 1st root of 1st order spherical Bessel function's derivative E(a) : weighting function, can be obtained via simulations S0 : Non diffusion-weighted signal D0 : diffusion coefficient
0.3
ΔS 0.2 0.1 10
30 tm / ms
50
Spherical pores of fixed radius can be measured using an exponential fit with parameter ω = α2D0/a2 and variable tm. For a mixture of different radii, the inverse Laplace transform yields a volume fraction distribution. Corresponding Author: Vincent Méthot Institute of Medical Engineering, University of Lübeck Ratzeburger Allee 160, 23562 Lübeck, Germany [email protected]
350
a
5 µm radius 10 µm radius 15 µm radius 20 µm radius 25 µm radius
Monte Carlo simulated data
0.4 Signal reconstructed from the estimated pore size distribution
0.2 Bi exponential decay (expected signal) 0.05
0.8
b
1.2
0.10
0.15
D0 = 2e 3/ ¹m 2 s −1 α = 2.08
300
Expected radii
b
tm / s
0.20
Cartoon of the voxel
1.0 0.8
Estimated pore volume fraction
0.6 0.4 0.2
5 µm
0.0 100
20 µm
101
Spherical pore radius / µm
102
250 Theoretical relationship (according to [3]) 0.6
ω=
200 Exponential fit
ω / s-1
For spherical pores, the normalized signal can be approximated by [3]: (eq. 1)
Parallel - antiparallel signal difference Δ S / a.u.
Theory
0.6
1.4
Figure 2 shows the signal difference simulated for various pore radii. Figure 3 shows the estimated sphere radius distribution from a combination of ΔS obtained with a = 5 µm and a = 20 µm pores. 1.0
a
0.8
0.0 0.00
Volume fraction [a. u.]
δ
time
Methods
1.0
Signal difference Δ S
Introduction
Δ S ∝ e −ω t m
0.4
Figure 3: (a) Sum of simulated ΔS for 5 µm and 20 µm spherical pore radii as well as theoretical and fitted decays. (b) solution of the Laplace inversion for the simulated data.
α2 D0 a2
150
Discussion
Monte Carlo simulated data 100
The simulated data exhibit the expected exponential decay with t m. Furthermore, it is possible to use this mixing-time dependency to retrieve information about a distribution of spherical pore radii by inverse Laplace transform.
Fitted ω from Figure 2a 0.2 50
0.0
0 0
50
100
Mixing time tm / ms
150
200
0
5
10
15
20
25
30
Pore radius (a) / µm
Figure 2: (a) Simulated ΔS for various spherical pore size radii and corresponding exponential fit. (b) Fitted decay parameter ω (from (a)) compared to theoretical prediction as in [3] (see eq. 2).
Funding: German Research Foundation, grant KO3389/2-1
References: [1] P. P. Mitra. Phys. Rev. B 51:15074–15078 (1995). [4] M. A. Koch and J. Finsterbusch. Magn. Reson. Med. 62:247–254 (2009). [2] D. Benjamini and U. Nevo. J. Magn. Reson. 230:198–204 (2013). [5] S. van der Walt et al. Comput. Sci. Eng. 13:22-30 (2011). [3] E. Özarslan and P. J. Basser. J. Chem. Phys. 128:154511 (2008).
