Publication detail

# Effects of axonal spatial distribution and diameter on diffusion MR simulations

LABOUNEK, R. MIKL, M. JAN, J. VALLA, R. BAŠTINEC, J. LENGLET, C.

Original Title

Effects of axonal spatial distribution and diameter on diffusion MR simulations

English Title

Effects of axonal spatial distribution and diameter on diffusion MR simulations

Type

abstract

Language

en

Original Abstract

Introduction: Since the introduction of diffusion tensor imaging [1], more complex models have been derived including some capable of estimating axonal diameter, density and other biophysical parameters [2-6]. One limitation of the models is that they may over-estimate diameters [5]. Although several complex Monte-Carlo (MC) simulators have been introduced [7-9], we present an intra-axonal compartment MC simulation framework, with lower computational requirements. It is due to the fact that, for a given diameter and fiber orientation, the signal from a single simulated fiber is sufficient to approximate the overall signal from a fiber population. We also show that fiber diameter causes characteristic changes in ratio between parallel and perpendicular diffusion MRI (dMRI) signal. Methods: For Brownian motion simulation inside geometrically constrained space, we have modified J. Burkardt's algorithm [10] for free diffusion with reflections on the medium boundary and without energy loss or elasticity as in [7]. The axon is approximated as cylinder. Temporal phase change for each water molecule is calculated with eq. (1), following [7]. dФ(n,t) is the temporal phase change at time t for n-th molecule, G(t) is diffusion gradient strength and direction, x(t) the molecule's position, γ the gyromagnetic ratio, and a is +1 during 1st gradient and -1 during 2nd gradient. Temporal increment dt is constant value. dΦ(n,t)=aγG(t)⋅x(t)dt (1) Final phase value for n-th water molecule is calculated with eq. (2). Φ(n)=∑d Φ(n,t) (2) Final dMRI signal S/S 0 for a given gradient is calculated with eq. (3). S/S 0 =(∑cos (Φ(n)))/N (3) First, we investigated the effect of spatial sampling of axons on dMRI signal (see Fig. 1). Fig. 1 (top) shows the final positions of molecules in 441 cylinders with diameter 8μm, placed uniformly in space. In Fig. 1 (bottom), the 441 cylinder occupy the same location L. Cylinders length was 1mm, diffusion constant 2*10 -9 m 2 /s, gradient strengths 10,20,30,...,110,120mT/m, and 642 gradient directions were used. Rectangular gradients had duration δ=20ms. Duration between gradients Δ was 30ms, and dt=2*10 -5 s. Initial molecule positions were uniformly distributed on 3D Cartesian grid with distances 1μm. Second, we investigated the effect of axon diameter on dMRI signal. We calculated dMRI signal for cylinder with length 300μm long placed at location L as shown in Fig.1 (bottom), with diameters varying from 0.4μm to 20μm with sampling step 0.4μm. Diffusion constant, gradient strengths and directions, δ and Δ were the same as in previous setting. Initial positions were sampled with distances from 0.2μm to 0.8μm depending on axon diameter and computing memory requirements. Results: In Fig. 2, for chosen directions from 441 phase distributions, the final average (μ) of phase (Φ) standard deviation (σ) μ σΦ was found to be the same for both axonal spatial distributions. The standard deviation of the phase standard deviations σ σΦ from 441 different axons was lower than differences between averages of the phase standard deviations μ σΦ across different gradient directions. In Fig. 3, normalized dMRI signal is shown for different gradient strengths. Ratio between perpendicular and parallel signals increases exponentially. Diameter effect results are shown in Fig. 4. Ratio between perpendicular and parallel signals is dependent on gradient strength and axonal diameter. For diameters higher than 3μm, the ratio decreases with increasing diameter at fixed gradient strength. Parallel dMRI signals appear to be independent of axon diameter. Conclusions: Simulations within a single cylinder (axon) appear sufficient to infer the dMRI signal originating from a population of spatially distributed and identical cylinders. Ratio between perpendicular and parallel dMRI signals may be a useful measure to estimate axonal diameters (larger than 3μm).

English abstract

Introduction: Since the introduction of diffusion tensor imaging [1], more complex models have been derived including some capable of estimating axonal diameter, density and other biophysical parameters [2-6]. One limitation of the models is that they may over-estimate diameters [5]. Although several complex Monte-Carlo (MC) simulators have been introduced [7-9], we present an intra-axonal compartment MC simulation framework, with lower computational requirements. It is due to the fact that, for a given diameter and fiber orientation, the signal from a single simulated fiber is sufficient to approximate the overall signal from a fiber population. We also show that fiber diameter causes characteristic changes in ratio between parallel and perpendicular diffusion MRI (dMRI) signal. Methods: For Brownian motion simulation inside geometrically constrained space, we have modified J. Burkardt's algorithm [10] for free diffusion with reflections on the medium boundary and without energy loss or elasticity as in [7]. The axon is approximated as cylinder. Temporal phase change for each water molecule is calculated with eq. (1), following [7]. dФ(n,t) is the temporal phase change at time t for n-th molecule, G(t) is diffusion gradient strength and direction, x(t) the molecule's position, γ the gyromagnetic ratio, and a is +1 during 1st gradient and -1 during 2nd gradient. Temporal increment dt is constant value. dΦ(n,t)=aγG(t)⋅x(t)dt (1) Final phase value for n-th water molecule is calculated with eq. (2). Φ(n)=∑d Φ(n,t) (2) Final dMRI signal S/S 0 for a given gradient is calculated with eq. (3). S/S 0 =(∑cos (Φ(n)))/N (3) First, we investigated the effect of spatial sampling of axons on dMRI signal (see Fig. 1). Fig. 1 (top) shows the final positions of molecules in 441 cylinders with diameter 8μm, placed uniformly in space. In Fig. 1 (bottom), the 441 cylinder occupy the same location L. Cylinders length was 1mm, diffusion constant 2*10 -9 m 2 /s, gradient strengths 10,20,30,...,110,120mT/m, and 642 gradient directions were used. Rectangular gradients had duration δ=20ms. Duration between gradients Δ was 30ms, and dt=2*10 -5 s. Initial molecule positions were uniformly distributed on 3D Cartesian grid with distances 1μm. Second, we investigated the effect of axon diameter on dMRI signal. We calculated dMRI signal for cylinder with length 300μm long placed at location L as shown in Fig.1 (bottom), with diameters varying from 0.4μm to 20μm with sampling step 0.4μm. Diffusion constant, gradient strengths and directions, δ and Δ were the same as in previous setting. Initial positions were sampled with distances from 0.2μm to 0.8μm depending on axon diameter and computing memory requirements. Results: In Fig. 2, for chosen directions from 441 phase distributions, the final average (μ) of phase (Φ) standard deviation (σ) μ σΦ was found to be the same for both axonal spatial distributions. The standard deviation of the phase standard deviations σ σΦ from 441 different axons was lower than differences between averages of the phase standard deviations μ σΦ across different gradient directions. In Fig. 3, normalized dMRI signal is shown for different gradient strengths. Ratio between perpendicular and parallel signals increases exponentially. Diameter effect results are shown in Fig. 4. Ratio between perpendicular and parallel signals is dependent on gradient strength and axonal diameter. For diameters higher than 3μm, the ratio decreases with increasing diameter at fixed gradient strength. Parallel dMRI signals appear to be independent of axon diameter. Conclusions: Simulations within a single cylinder (axon) appear sufficient to infer the dMRI signal originating from a population of spatially distributed and identical cylinders. Ratio between perpendicular and parallel dMRI signals may be a useful measure to estimate axonal diameters (larger than 3μm).

Keywords

MRI, white matter imaging, DTI, HARDI, DSI

Released

26.06.2016

Publisher

OHBM

Location

Geneva

Pages from

1

Pages to

5

Pages count

5

URL