Diffusion time dependency of extracellular diffusion

Abstract

Purpose: To quantify the variations of the power-law dependences on diffusion time t or gradient frequency $f$ of extracellular water diffusion measured by diffusion MRI (dMRI).

Methods: Model cellular systems containing only extracellular water were used to investigate the $t/f$ dependence of $D_{\mathrm{ex}}$ , the extracellular diffusion coefficient. Computer simulations used a randomly packed tissue model with realistic intracellular volume fractions and cell sizes. DMRI measurements were performed on samples consisting of liposomes containing heavy water(D₂ O, deuterium oxide) dispersed in regular water (H₂ O). $D_{\mathrm{ex}}$ was obtained over a broad $t$ range (∼1-1000 ms) and then fit power-law equations $D_{\mathrm{ex}}\left(t\right)=D_{\text{const}}+\text{const}·t^{-{\vartheta }_t}$ and $D_{\mathrm{ex}}\left(f\right)=D_{\text{const}}+\text{const}·f^{{\vartheta }_f}$ .

Results: Both simulated and experimental results suggest that no single power-law adequately describes the behavior of $D_{\mathrm{ex}}$ over the range of diffusion times of most interest in practical dMRI. Previous theoretical predictions are accurate over only limited $t$ ranges; for example, ${\theta }_t={\theta }_f=-\frac{1}{2}$ is valid only for short times, whereas ${\theta }_t=1$ or ${\theta }_f=\frac{3}{2}$ is valid only for long times but cannot describe other ranges simultaneously. For the specific $t$ range of 5-70 ms used in typical human dMRI measurements, ${\theta }_t={\theta }_f=1$ matches the data well empirically.

Conclusion: The optimal power-law fit of extracellular diffusion varies with diffusion time. The dependency obtained at short or long $t$ limits cannot be applied to typical dMRI measurements in human cancer or liver. It is essential to determine the appropriate diffusion time range when modeling extracellular diffusion in dMRI-based quantitative microstructural imaging.

Keywords: D2O; diffusion; diffusion time; extracellular; liposome; oscillating gradient; phantom.

Figure 1

A representative distribution of cell radii (A) and the diagram of the corresponding randomly-packed tissue model used in the simulations (B) with an intracellular volume fraction = 75%.

Figure 2

Diagram of the liposome model system. All liposome enclosed space is occupied by heavy water (D₂O) that does not contribute to proton dMRI signals. Hence, measurements are determined by interstitial extracellular regular water (H₂O) only.

Figure 3

Simulated extracellular diffusion coefficient $D_{ex}$ dependent on diffusion time $t$ and gradient frequency $f$ (markers) and the corresponding fitted curves with typical power-law exponents in different $t/f$ ranges. The shaded area indicates the typical $t/f$ ranges available on clinical MRI systems. A base-10 logarithmic scale is used on the $t$ axis.

Figure 4

MR measured extracellular diffusion coefficient $D_{ex}$ of six liposome samples dependent on diffusion time $t$ (A) and gradient frequency $f$ (B). The shaded area indicates the typical $t$ range 5 – 70 ms and $f$ range $f$ < 50 Hz used on clinical MRI systems. A base-10 logarithmic scale is used on all $t$ axes.

Figure 5

NRMSE (normalized root mean square error) dependent on power-law exponents in simulations (A) and liposome experiments (B). Note that ${\vartheta }_f$ can only be zero or positive because PGSE $f$ is assumed to be 0.