TY - JOUR

T1 - Using spectral and cumulative spectral entropy to classify anomalous diffusion in Sephadex™ gels

AU - Liang, Yingjie

AU - Chen, Wen

AU - Akpa, Belinda S.

AU - Neuberger, Thomas

AU - Webb, Andrew G.

AU - Magin, Richard L.

N1 - Funding Information:
The work described in this paper was supported by the National Science Funds for Distinguished Young Scholars of China (11125208), the 111 project under Grant B12032, the China Scholarship Council (CSC) (Grant No. 201406710047), and the Hohai University Training Program for Excellent Doctoral Dissertations (Grant No. 2013B15020056).
Publisher Copyright:
© 2017 Elsevier Ltd

PY - 2017/3/1

Y1 - 2017/3/1

N2 - Sephadex™ gel beads are commonly used to separate mixtures of similar molecules based on trapping and size exclusion from internal submicron diameter cavities. Water, as it freely moves through the porous gel and enclosed chambers of Sephadex™ beads, exhibits both normal (Gaussian) and anomalous (non-Gaussian) water diffusion. The apparent diffusion coefficient (ADC) of water in Sephadex™ gels can be measured using magnetic resonance imaging (MRI) by applying diffusion-weighted pulse sequences. This study investigates the relationship between the ADC of water and the complexity (i.e., size and number of cavities) of a series of Sephadex™ beads. We first classified the stochastic movement of water by using the solution to the space and time fractional diffusion equation to extract the ADC and the fractional time and space parameters (α, β), which are essentially the order of the respective fractional derivatives in Fick's second law. From the perspective of the continuous time random walk (CTRW) model of anomalous diffusion, these parameters reflect waiting times (trapping) and jump increments (nano-flow) of the water in the gels. The observed MRI diffusion signal decay represents the Fourier transform of the diffusion propagator (i.e., the characteristic function of the stochastic process). In two series of Sephadex™ gel beads, we observed a strong inverse correlation between bead porosity (which is also responsible for molecular size exclusion) and the fractional order parameters; as the gels become more heterogeneous, the ADC decreases, both α and β are reduced and the diffusion exhibits anomalous (sub-diffusion) behavior. In addition, as a new measure for the structural complexity in Sephadex™ gel beads, we propose using the spectral and the cumulative spectral entropy that are derived from the observed characteristic function. We find that both measures of entropy increase with the porosity and tortuosity of the gel in a manner consistent with fractional order diffusional dynamics.

AB - Sephadex™ gel beads are commonly used to separate mixtures of similar molecules based on trapping and size exclusion from internal submicron diameter cavities. Water, as it freely moves through the porous gel and enclosed chambers of Sephadex™ beads, exhibits both normal (Gaussian) and anomalous (non-Gaussian) water diffusion. The apparent diffusion coefficient (ADC) of water in Sephadex™ gels can be measured using magnetic resonance imaging (MRI) by applying diffusion-weighted pulse sequences. This study investigates the relationship between the ADC of water and the complexity (i.e., size and number of cavities) of a series of Sephadex™ beads. We first classified the stochastic movement of water by using the solution to the space and time fractional diffusion equation to extract the ADC and the fractional time and space parameters (α, β), which are essentially the order of the respective fractional derivatives in Fick's second law. From the perspective of the continuous time random walk (CTRW) model of anomalous diffusion, these parameters reflect waiting times (trapping) and jump increments (nano-flow) of the water in the gels. The observed MRI diffusion signal decay represents the Fourier transform of the diffusion propagator (i.e., the characteristic function of the stochastic process). In two series of Sephadex™ gel beads, we observed a strong inverse correlation between bead porosity (which is also responsible for molecular size exclusion) and the fractional order parameters; as the gels become more heterogeneous, the ADC decreases, both α and β are reduced and the diffusion exhibits anomalous (sub-diffusion) behavior. In addition, as a new measure for the structural complexity in Sephadex™ gel beads, we propose using the spectral and the cumulative spectral entropy that are derived from the observed characteristic function. We find that both measures of entropy increase with the porosity and tortuosity of the gel in a manner consistent with fractional order diffusional dynamics.

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U2 - 10.1016/j.camwa.2016.12.028

DO - 10.1016/j.camwa.2016.12.028

M3 - Article

AN - SCOPUS:85010202543

SN - 0898-1221

VL - 73

SP - 765

EP - 774

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 5

ER -