Mean Kärger Model Water Exchange Rate in Brain

Abstract

Intercellular water exchange in brain is analyzed in terms of the multi-compartment Kärger model (KM), and the mean KM water exchange rate is used as a summary statistic for characterizing the exchange processes. Prior work is extended by deriving a stronger lower bound for mean exchange rate that can be determined from the time dependence of the diffusional kurtosis. In addition, an analytic formula giving the time dependence of the kurtosis for a model of thin cylindrical neurites is demonstrated, and this formula is applied to numerically test the accuracy of the lower bound for a range of model parameters. Finally, the lower bound is measured in vivo with diffusional kurtosis imaging for the dorsal hippocampus and cerebral cortex of eight-month-old mice. From the stronger lower bound, the mean KM exchange rate is found to be $46.1\pm 11.0s^{-1}$ or greater in dorsal hippocampus and $20.5\pm 8.5s^{-1}$ or greater in cortex.

Keywords: Kärger model; brain; diffusion; kurtosis; mouse; water exchange.

Fig. 1.

(A) The function$V(x)$needed to calculate the enhancement factor$E_f$. The solid line shows the exact form given byEquations 11and12while the dotted line is the sixth order approximation ofEquation 13.$V(x)$is only defined for$0\le x<3$, which is the range of physical interest. (B) Ratio$R_{KM}/R_{\text{in}}$as a function of${\kappa }_N/K_0$for several values of the extra-neurite water fraction$f_{ex}$in the thin cylindrical neurite model. If${\kappa }_N/K_0\ll f_{\text{ex}}/(1-f_{\text{ex}})$, then$R_{KM}\approx R_{\text{in}}$, but$R_{KM}$may be substantially larger than$R_{\text{in}}$otherwise. However,$R_{\text{in}}\le R_{KM}\le R_{\text{in}}/f_{\text{ex}}$in all cases.

Fig. 2.

Fractional anisotropy (FA) map from one animal illustrating the two regions of interest (ROIs) used in this study for the dorsal hippocampus (outlined in orange) and cortex (outlined in blue). For each animal, data were pooled from both hemispheres of the brain within a single coronal slice.

Fig. 3.

Accuracy of the lower bound$R_{KM}^{*}$expressed as the ratio$R_{KM}^{*}/R_{KM}$as a function of$R_{KM}^{*}{t}^{*}$in the thin cylindrical neurite model (A-D). For the cases considered, the accuracy is better than 73% for$R_{KM}^{*}{t}^{*}\le 0.5$and better than 51% for$R_{KM}^{*}{t}^{*}\le 1.0$. The accuracy becomes less sensitive to${\kappa }_N/K_0$as$f_{\text{ex}}$is increased. The accuracies with${\kappa }_N/K_0=0$and${\kappa }_N/K_0=1$are identical and independent of$f_{\text{ex}}$.

Fig. 4.

Accuracy of the lower bound${\widehat{R}}_{KM}$expressed as the ratio${\widehat{R}}_{KM}/R_{KM}$as a function of$R_{KM}^{*}{t}^{*}$in the thin cylindrical neurite model (A-D). For the cases considered, the accuracy is better than 80% for$R_{KM}^{*}{t}^{*}\le$0.5 and better than 63% for$R_{KM}^{*}{t}^{*}\le$1.0. The accuracies are 100% with${\kappa }_N/K_0=$0 and${\kappa }_N/K_0=$ 1 for all values of$f_{ex}$. The lower bound${\widehat{R}}_{KM}$is always more accurate than$R_{KM}^{*}$.

Fig. 5.

The mean diffusivity as a function of diffusion time for the dorsal hippocampus (DH) and the cortex (CX). Data from the normal control (NC) mice are shown in panels (A) and (C), while data from the transgenic (TG) mice are shown in panels (B) and (D). The data points show mean values for individual animals, with distinct symbols distinguishing animals within a group. Linear least-squares fits (solid lines) indicate that the diffusivity varies little over the time range considered and is similar across regions and groups.

Fig. 6.

Log-log plot of the same data shown inFigure 5(A-D). The slopes of the linear fits (solid lines) give the diffusion elasticity$\xi$, which provides a quantitative criterion for assessing the strength of the diffusivity’s time dependence. In all cases, the elasticity has a magnitude much less than one implying that the mean diffusivity can be considered to be approximately constant for diffusion times between 18 and 30 ms in consistency with the KM.

Fig. 7.

The mean kurtosis as a function of diffusion time for the two ROIs. The data from the NC mice are shown in panels (A) and (C), and the data from the TG mice are shown in panels (B) and (D). Linear fits (solid lines) indicate that the kurtosis decreases with increasing diffusion time.

Fig. 8.

Semi-log plot of the same data shown inFigure 7(A-D). The slopes of the linear fits (solid lines) times -3 give the lower bound$R_{KM}^{*}$for the mean KM exchange rate. The values for$R_{KM}^{*}$are similar for NC and TG mice.

Fig. 9.

Myelin basic protein (MBP) stain (1.25× magnification) of a TG mouse obtained with immunohistochemistry (IHC). The optical density (OD) is several times higher in CX compared to the DH indicating a substantially greater degree of myelination in CX. This difference could contribute to the observed higher$R_{KM}^{*}$values for DH since water exchange is expected to be slower for myelinated axons.