# Diffusion in Sephadex Gel Structures: Time Dependency Revealed by Multi-Sequence Acquisition over a Broad Diffusion Time Range

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## Abstract

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## 1. Introduction

_{eff}), which is constrained by the separation and duration of diffusion gradient lobes. For water molecules that diffuse during Δ

_{eff}, the varying degree of spatial dislocation results in a phase dispersion (Φ) of the magnetization. The probabilistic distribution function (PDF) of the net displacement of diffusing water molecules is related to a probability distribution of Φ, which leads to signal attenuation in an MRI measurement [3,4].

^{2}〉 ~ t [5]. In the presence of restricting barriers in complex materials, however, the probability distribution of molecular displacement no longer follows Gaussian distribution. The MSD in the non-Gaussian diffusion case can be characterized as a function of intrinsic diffusion coefficient, restrictive geometry, and diffusion time [6]. One way of characterizing the non-Gaussian diffusion behavior is to employ the continuous time random walk (CTRW) theory, in which the MSD can be expressed by a composite power law as: 〈x

^{2}〉 ~ t

^{2α/β}, where α and β are the fractional order time and space derivatives, respectively, in fractionalized Fick’s second law [7,8]. This generalized description enables the CTRW model to provide a more realistic description of the complex diffusion pattern in biological tissues [8].

## 2. Theory

^{2}(t)〉 ~ t [8].

_{α,β}is the anomalous diffusion coefficient (in mm

^{β}/s

^{α}), ${}_{0}{}^{C}D{}_{t}^{\alpha}$ is the αth (0 < α ≤ 1) fractional order time derivative in the Caputo form, given as [21]:

^{2}(t)〉 ~ t

^{(2α/β)}. When α = 1 and β = 2, this formalism is reduced to the classical Gaussian expression. In comparison, when 2α > β or 2α < β, the anomalous diffusion process is referred to as super-diffusive or sub-diffusive [7,17] respectively, and when 2α = β, the non-Gaussian dynamics is described as quasi-diffusion [23].

_{0}is the signal intensity without diffusion weighting and S is the signal intensity at q and Δ

_{eff}, where q = γG

_{diff}δ and Δ

_{eff}= Δ − δ/3 in which γ is the gyrometric ratio, G

_{diff}is the diffusion gradient amplitude, δ is the diffusion gradient pulse width, and Δ is the diffusion gradient separation. E

_{α}is a single-parameter Mittag–Leffler function [8]. For other diffusion gradient waveforms, expressions analogous to Equation (7) can be derived in reference of the methods described in [24,25].

## 3. Methods

#### 3.1. Sephadex Gel Phantom Preparation

#### 3.2. Data Acquisition

_{eff}and b-value of the cosine-trapezoid OGSE sequence are given by [26]:

_{eff}= δ/(3N)

_{eff}values of 1.67, 2.5, and 5 ms, respectively.

_{eff}, under a Stejskal–Tanner diffusion sensitizing gradient pair in a PGSE sequence is given by [3]:

_{eff}= Δ − δ/3

_{eff}values of 10.17 ms and 34.17 ms, respectively (Figure 1b).

_{eff}values were 34.17, 59.17, 99.17, and 149.17 ms, respectively (Figure 1c).

^{2}) were acquired from the Sephadex gel phantoms by varying G

_{diff}. The other imaging parameters, TR (4000 ms) and TE (75 ms), diffusion gradient direction = R/L, FOV = 36 × 36 mm

^{2}, acquisition matrix = 32 × 32, slice thickness = 2 mm and number of repetitions (NEX = 4), were kept the same in all sequences.

#### 3.3. Data Analysis

_{0}. The CTRW model in Equation (7) was fit to the DW images voxel-by-voxel by using an iterative non-linear Levenberg–Marquardt algorithm in MATLAB. To improve the fitting stability and alleviate the degeneracy, D

_{α,β}was first estimated by a mono-exponential model at lower b-values, followed by a simultaneous estimation of other parameters with appropriate constraints (0 < α ≤ 1 and 0 < β ≤ 2) by using all b-values [27]. Measurements were made from each quantitative parameter map (D

_{α,β}, α, and β) by computing the mean value over a ~16 mm

^{2}region-of-interest (ROI) within each vial of the Sephadex gel. Representative DW images and ROIs are shown in Figure 2.

#### 3.4. Monte Carlo Simulations

_{eff}values (3.33, 5, and 10 ms). Eleven different synthetic PGSE/PGSTE signals were simulated with Stejskal–Tanner diffusion gradients at eleven Δ

_{eff}(25, 30, 35, 40, 45, 50, 60, 70, 80, 90, and 100 ms).

^{−3}mm

^{2}/s, intracellular volume ratio of 0.5 and 7 b-values (0, 200, 500, 1000, 1500, 3000, and 6000 s/mm

^{2}). The normalized simulated signal intensities over all b-values were fit to Equation (7) with the same fitting procedure as for the experimental data.

## 4. Results

_{α,β}, obtained from the first Sephadex gel series was plotted as a function of Δ

_{eff}in Figure 3. Two trends were observed. First, a downward trend reaching a plateau was seen for all gels, suggesting increased hinderance at longer diffusion times. Second, the gels with larger pore sizes (G50–50 and G75–50 in Figure 3b,c) exhibited higher D

_{α,β}values at all diffusion times.

_{eff}, respectively. α and β exhibited a similar trend to each other. For all the gels, as Δ

_{eff}decreased to 0, α and β values approached to 1 and 2, respectively, indicating that the diffusion signal behavior approaches to the Gaussian regime in the limit of short diffusion times. The gels with larger pore sizes exhibited higher α and β values (G50–50 and G75–50 in Figure 4a,b and Figure 5a,b), suggesting less deviation from Gaussian diffusion dynamics.

_{α,β}is plotted as a function of Δ

_{eff}for the second Sephadex gel series, G50–50, G50–80, and G50–150. Similar to the first Sephadex gel series, D

_{α,β}followed a downward trend reaching a plateau for all the Sephadex gels. G50–50, G50–80, and G50–150 in Figure 6a–c show similar D

_{α,β}values at short diffusion times. However, at longer diffusion times, the gels with larger bead sizes (G50–80 and G50–150 in Figure 6b,c) exhibited higher D

_{α,β}values, similar to what is shown in Figure 3.

_{eff}for gels in the second Sephadex series (G50–50, G50–80, and G50–150). Similar to the first Sephadex gel series, α and β showed a decreasing trend against Δ

_{eff}in all gels. As the dry bead size increased, higher α and β values were observed in general. 2α/β < 1 was observed in all Sephadex gels, indicating the diffusion dynamics fell into sub-diffusion regime.

_{α,β}(Figure 9a), α (Figure 9b), and β (Figure 9c), yielded higher values in the data with higher permeability. In the second simulation dataset (fixed p and varying r of 6, 7, and 8 µm), the simulated data with a larger cylinder radius yielded higher D

_{α,β}(Figure 10a), α (Figure 10b), and β (Figure 10c). The simulation results exhibited a good agreement with experimental results. In both simulations, D

_{α,β}, α, and β exhibited a monotonically decreasing trend.

## 5. Discussion

_{α,β}, α, and β, as the diffusion time increased. Our Monte Carlo simulations exhibited a similar trend with the experimental results. To the best of our knowledge, this is the first study which investigates the time dependency behavior of the CTRW model over a wide range of diffusion times using a multi-sequence acquisition scheme.

_{α,β}converged to the diffusion coefficient of pure water, D

_{0}. These outcomes were clearly observed in our experimental data and confirmed in Monte Carlo simulations. In contrast, the long diffusion time provides water molecules a greater opportunity to explore the heterogeneity of the surrounding environment, resulting in reduced α and β. The increased hinderance and restriction experienced by water molecules at long diffusion times yielded reduced D

_{α,β}, α, and β values, which allows us to infer information on microstructures and micro-environment.

_{α,β}values observed in the gels with smaller pore sizes in the first Sephadex gel series (Figure 3a,b) is consistent with the general belief that diffusion coefficient is lower in materials with increased micro-structural barriers [31]. Sephadex gels with larger dry bead sizes exhibited similar D

_{α,β}at low diffusion times, but higher D

_{α,β}at long diffusion times (Figure 6b,c), suggesting that the influence of microstructure scale on diffusion dynamics is more evident at longer diffusion times.

_{α,β}values obtained from the experiments performed with the PGSTE were higher than those observed by the PGSE at Δ

_{eff}= 35 ms (Figure 3 and Figure 6). Also, α and β values estimated from the data acquired with the PGSE at Δ = 11 ms were higher than those with the OGSE at Δ

_{eff}= 5 ms in the first Sephadex series, as shown in Figure 4 and Figure 5. The root cause of the discontinuities is unknown and requires further investigation. Nevertheless, the overall monotonic trend across a broad diffusion time range is consistent in all CTRW parameters. Two Sephadex gel series with varying pore size or bead diameter exhibited the same monotonic trend, which was consistent with the trends revealed by the Monte Carlo simulations.

_{eff}in our experiments was limited to 149.17 ms. This was largely due to the inadequate signal-to-noise ratio in the PGSTE acquisition. Additionally, a moderate TE of 75 ms was chosen to match the TE in the OGSE acquisition, thereby mitigating the potential issue with the TE-dependence in diffusion characterization, which can be particularly evident in a multi-compartmental environment [36,37,38]. If the PGSTE sequence is employed alone without the need to match parameters in other sequences, then studies on diffusion time dependency at long diffusion time regime can be conducted with a shorter TE. Second, our simulations did not cover the low diffusion time regime (e.g., <3.33 ms for OGSE and <25 ms for PGSE/PGSTE). This was because the limited total step size of 20,000 could lead to unstable results at shorter diffusion times. Optimized algorithms and more powerful computational platforms may help overcome this limitation. Third, although significant time dependency of CTRW model parameters was observed in this study, the analytical expressions [39] of this time dependency in a multi-compartmental environment requires further investigation. Finally, Sephadex gel phantoms provide a simple diffusion environment with spherical beads and permeable pores. Although they helped provide valuable insights into understanding of the complex diffusion processes, their limitations in adequately mimicking actual biological tissue structures must be recognized.

## 6. Conclusions

_{α,β}, α, and β as the diffusion time increased. These experimental results were reinforced by the Monte Carlo simulations. The present study provides valuable insights into probing microstructures by characterizing the time dependency of the CTRW model parameters, paving the way for future investigations on biological systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Pulse sequences employed in this study. (

**a**) Cosine-trapezoid OGSE where N is the number of half oscillation period and δ is the total waveform duration (N = 4 in the sequence diagram). (

**b**) PGSE where δ is the diffusion lobe duration and Δ is the diffusion lobe separation. (

**c**) PGSTE where δ and Δ are defined similarly as in (

**b**).

**Figure 2.**DW images acquired by using the PGSE sequence (Δ = 35 ms and b = 0). (

**a**) The first Sephadex series: G25–50 (bottom right), G50–50 (bottom left), and G75–50 (top). (

**b**) The second Sephadex series: G50–50 (bottom left), G50–80 (bottom right), and G50–150 (top). The rhombus-shaped ROIs indicate the regions used to calculate the mean parameter values.

**Figure 3.**Plots of D

_{α,β}versus effective diffusion time, Δ

_{eff}, for gels G25–50 (

**a**) and G50–50 (

**b**), and G75–50 (

**c**) with increased macromolecular exclusion limit. The data acquired by using the OGSE, PGSE, and PGSTE pulse sequences are marked in black, red, and blue, respectively.

**Figure 4.**Plots of temporal fractional order (α) versus effective diffusion time, Δ

_{eff}, for gels G25–50 (

**a**) and G50–50 (

**b**), and G75–50 (

**c**). The data acquired by using the OGSE, PGSE, and PGSTE pulse sequences are marked in black, red, and blue, respectively.

**Figure 5.**Plots of spatial fractional order (β) versus effective diffusion time, Δ

_{eff}, for gels G25–50 (

**a**) and G50–50 (

**b**), and G75–50 (

**c**). The data acquired by using the OGSE, PGSE, and PGSTE pulse sequences are marked in black, red, and blue, respectively.

**Figure 6.**Plot of D

_{α,β}versus effective diffusion time, Δ

_{eff}, for gels G50–50 (

**a**), G50–80 (

**b**), and G50–150 (

**c**). The data acquired by using the OGSE, PGSE, and PGSTE pulse sequences are marked in black, red, and blue, respectively.

**Figure 7.**Plot of temporal fractional order (α) versus effective diffusion time, Δ

_{eff}, for gels G50–50 (

**a**), G50–80 (

**b**), and G50–150 (

**c**). The data acquired by using the OGSE, PGSE, and PGSTE pulse sequences are marked in black, red, and blue, respectively.

**Figure 8.**Plot of spatial fractional order (β) versus effective diffusion time, Δ

_{eff}, for gels G50–50 (

**a**), G50–80 (

**b**), and G50–150 (

**c**). The data acquired by using the OGSE, PGSE, and PGSTE pulse sequences are marked in black, red, and blue, respectively.

**Figure 9.**Plots of D

_{α,β}(

**a**), α (

**b**), and β (

**c**) versus Δ

_{eff}obtained from the Monte Carlo simulations with fixed r = 8 µm and varying p of 0.1% (red), 0.2% (green) and 0.4% (blue). The rhombi and circles represent the simulation results with oscillating diffusion gradient (OGSE) and Stejskal–Tanner diffusion gradient (PGSE/PGSTE), respectively.

**Figure 10.**Plots of D

_{α,β}(

**a**), α (

**b**), and β (

**c**) versus Δ

_{eff}from Monte Carlo simulation with fixed p = 0.2% and varying r of 6 µm (red), 7 µm (green) and 8 µm (blue). The rhombi and circles represent simulation results with oscillating diffusion gradient (OGSE) and Stejskal–Tanner diffusion gradient (PGSE/PGSTE), respectively.

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**MDPI and ACS Style**

Dan, G.; Li, W.; Zhong, Z.; Sun, K.; Luo, Q.; Magin, R.L.; Zhou, X.J.; Karaman, M.M. Diffusion in Sephadex Gel Structures: Time Dependency Revealed by Multi-Sequence Acquisition over a Broad Diffusion Time Range. *Mathematics* **2021**, *9*, 1688.
https://doi.org/10.3390/math9141688

**AMA Style**

Dan G, Li W, Zhong Z, Sun K, Luo Q, Magin RL, Zhou XJ, Karaman MM. Diffusion in Sephadex Gel Structures: Time Dependency Revealed by Multi-Sequence Acquisition over a Broad Diffusion Time Range. *Mathematics*. 2021; 9(14):1688.
https://doi.org/10.3390/math9141688

**Chicago/Turabian Style**

Dan, Guangyu, Weiguo Li, Zheng Zhong, Kaibao Sun, Qingfei Luo, Richard L. Magin, Xiaohong Joe Zhou, and M. Muge Karaman. 2021. "Diffusion in Sephadex Gel Structures: Time Dependency Revealed by Multi-Sequence Acquisition over a Broad Diffusion Time Range" *Mathematics* 9, no. 14: 1688.
https://doi.org/10.3390/math9141688