Apparent Exchange Rate (AXR) of water between micro-environments with different apparent diffusivities can be quantified by filter exchange imaging (FEXI). FEXI is based on a double diffusion encoding (DDE) experiment with a variable mixing time^1,2. In tissue, AXR may depend on the diffusion encoding direction³. Understanding the effect that diffusion anisotropy has on AXR is fundamental in experimental design and interpretation of results. We present illustrative examples of anisotropic conditions that affect AXR. This study indicates that AXR is expected to be confounded even in simple systems with a single exchange rate if there are more than two orientationally dispersed compartments.

Applying the original definition of AXR¹ to anisotropic systems, we have

$${\rm AXR}(\hat{\bf g})=-\lim_{t_{\rm m} \rightarrow 0}\frac{\partial }{\partial t_{\rm m}}⁡ \ln\left[\frac{D(\hat{\bf g})-D'(b_{\rm f},\hat{\bf g},t_{\rm m})}{D(\hat{\bf g})-D'(b_{\rm f},\hat{\bf g},t_{\rm m}=0)}\right],$$

where $$$\hat{\bf g}$$$ is the gradient encoding direction, t_m is the mixing time, b_f is the filter b-value and $$$D(\hat{\bf g})=\hat{\bf g}^T\bf{D}\hat{\bf g}$$$ and $$$D'(b_{\rm f},\hat{\bf g},t_{\rm m})=\hat{\bf g}^T {\bf D'}(b_{\rm f},t_{\rm m})\hat{\bf g}$$$ are projections along $$$\hat{\bf g}$$$ for the equilibrium and filtered diffusion tensors, respectively. The filter efficiency¹ also depends on direction and is given by

$$\sigma(\hat{\bf g})=1-\frac{D'(b_{\rm f},\hat{\bf g},t_{\rm m}=0)}{D(\hat{\bf g})}.$$

For n exchanging compartments, the equilibrium and filtered diffusivities are given by the weighted average $$D(\hat{\bf g})=\sum_n X_n^{\rm eq}D_n(\hat{\bf g})\;\;{\rm and}\\D'(b_{\rm f},\hat{\bf g},t_{\rm m})=\sum_n X_n(b_{\rm f},\hat{\bf g},t_{\rm m})\;D_n(\hat{\bf g}),$$

where $$$X_n^{\rm eq}$$$ and $$$X_n(b_{\rm f},\hat{\bf g},t_{\rm m})$$$ are signal population fractions for compartment n at equilibrium and after filtering, respectively. The fractions are perturbed depending on direction

$$X_n(b_{\rm f},\hat{\bf g},t_{\rm m}=0)=\frac{X_n^{\rm eq}\exp[-b_{\rm f}D_n(\hat{\bf g})]}{\sum_k X_k^{\rm eq}\exp[-b_{\rm f}D_k(\hat{\bf g})]}.$$

The evolution of the fractions $$$ {\bf X}(b_{\rm f},\hat{\bf g},t_{\rm m})$$$ follows first order exchange kinetics^4,5,

$${\bf X}(b_{\rm f},\hat{\bf g},t_{\rm m})=\exp({\bf K}t_{\rm m})\;{\bf X}(b_{\rm f},\hat{\bf g},t_{\rm m}=0),$$

where K is the rate matrix fulfilling the equilibrium condition $$${\bf K}\cdot{\bf X}^{\rm eq}= 0$$$.

Consider a case of three anisotropic components, two intracellular and one extracellular (Fig. 1A). Spins in the two intracellular compartments (equilibrium signal fractions f₁ and f₂,) can exchange between each other only via the extracellular compartment (signal fraction f_e). The rate matrix is given by

$${\bf K}=\begin{bmatrix}-k_{\rm 1e} & 0 & k_{\rm e1} \\0 & -k_{\rm 2e} & k_{\rm e2} \\k_{\rm 1e} & k_{\rm 2e} & -k_{\rm e1}-k_{\rm e2} \end{bmatrix}.$$

The $$${\rm AXR}(\hat{\bf g})$$$ was calculated analytically in the limit of low b_f (expression not reported here). For each compartment in our examples, we used axially symmetric diffusion tensors with axial and radial diffusivities given by: D₀ and D₀/10 for the extracellular (Fig. 1B,D,E), D₀/2 and D₀/20 for the intracellular (Fig. 1B,D,E) and D₀ for the isotropic extracellular compartment (Fig. 1C), where D₀ = 10^-9 m²s^-1. We set k_1e= k_2e = 1 s^-1. The signal fractions in equilibrium were set to X^eq = (0.8,0.2,0) (Fig. 1B) or X^eq = (0.4,0.2,0.4) (Fig. 1C,D). In Fig. 1E we used 1000 anisotropic intracellular compartments uniformly distributed in plane together amounting to 80% of the equilibrium signal fraction. For this example, the exchange matrix K is given in a similar fashion as above, but including 1000 intracellular compartments, and $$${\rm AXR}(\hat{\bf g})$$$ was numerically evaluated using b_f=10¹⁰sm^-2 and t_m time step of 1μs.

Our model system allows investigating effects of anisotropy on AXR. It mimics relevant tissue scenarios where water exchanges between intracellular and extracellular compartments. For systems with only one intracellular compartment (Fig. 1B), AXR = k_1e + k_e1, which is independent of direction, regardless of anisotropy of any of the compartments. The filter efficiency, σ, will in general depend on direction, which might introduce a fitting bias¹. In this case, it might be advantageous to fit the AXR model to arithmetically averaged data acquired in multiple directions², since this provides optimally maximized filtering. Alternatively, data from different directions could be fitted with a single AXR value using tensor representations for D, D’ and σ.

When more than two anisotropic compartments are included (Fig. 1C-E), the AXR depends on direction even in simple systems with a single exchange rate (intracellular lifetime), provided that intracellular compartments are orientationally dispersed. This effect can be understood by noting that AXR is reduced along the main axis of intracellular compartments (Fig. 1C-E). The AXR is dominated by exchange from those compartments that are mostly affected by the filter. Examples in Fig. 1C-E also show that the degree of AXR anisotropy cannot be inferred from the diffusion tensor model alone. Furthermore, the filter efficiency σ, which is very sensitive to anisotropy, cannot reveal the presence of multiple compartments. Higher specificity to different anisotropic configurations can thus be achieved if a combination of $$$D(\hat{\bf g})$$$, $$$\sigma(\hat{\bf g})$$$ and $$${\rm AXR}(\hat{\bf g})$$$ is considered, which may guide identification of different fiber populations and their directions within a voxel. In conclusion, we have found that the AXR is anisotropic even in simple systems. This is of importance in the interpretation of the AXR, but could also be useful for detailed analysis of fiber-specific characteristics.