The two-compartment free-water model accounts for partial volume effects that occur in diffusion tensor imaging (DTI) when image voxels contain both a tissue compartment and a fast diffusing free-water compartment [1]. Free-water is defined as water molecules that do not experience hindrance or restriction. Partial volume with free-water can be found near cerebrospinal fluid filled spaces such as the ventricles, or around the brain parenchyma but also to a lesser extent in deep white- and gray matter structures, where free-water is found in extracellular spaces [1].

The free-water model requires one additional parameter compared to the DTI model. This additional parameter complicates the model fit, requiring regularization [1] or multi-shell acquisitions [2] to arrive at a feasible solution. Here, we extend the work in [2] by exploring the solution space of the free-water problem both in single- and multi-shell contexts and under different noise levels. The purpose of this work is to explore the noise sensitivity of the free-water model fit, and investigate the advantages of using a multi-shell approach.

We consider the two-compartment free-water model (1)

$$s_i = s_0 [ (1-f) \exp (-b_i g_i^T D_{tissue} g_i) + f \exp( -b_i D_{free-water}) ]$$

where $$$f$$$ is the fractional volume of the free-water compartment, $$$D_{tissue}$$$ is the tensor of the tissue compartment, and $$$D_{free-water}=3\cdot 10^{-3}mm^2/s$$$. To explore the model we generate synthetic data with known optimal solutions, and systematically sample the solution space.

For our experiments we utilize (a) an optimized two-shell gradient scheme with $$$32+32$$$ directions, $$$b=\{500,1500\}s/mm^2$$$ [2] and (b) a single-shell scheme with 64 gradients, $$$b=1000s/mm^2$$$. Using eq. 1, we generate diffusion weighted images where we let $$$f=\frac{1}{2}$$$, and tensor eigenvalues $$$\lambda=\{1.4,0.4,0.3\}$$$ (anisotropic) or $$$\lambda=\{0.73,0.73,0.73\}$$$ (isotropic) as two separate test cases. Finally we add Rician noise with varying noise-levels.

The solution space is spanned by iteratively fixating different values of $$$f \in [0,1]$$$ and solving for $$$D_{tissue}$$$ using a weighted linear least squares (WLLS) method [2]. We report the residual of the estimated fits for various $$$f$$$ and noise-levels, and indicate where negative eigenvalues are detected, as upper bounds on the solution.

In figures 1-4, each curve represents a given noise level, and we log-plot the residual of the fit against different choices of $$$f$$$. The $$$f$$$ with minimal residual is marked in red. The first occurrence of negative eigenvalues is marked with a $$$\times$$$.

In figures 1-2 we validate that WLLS is not appropriate for single shell data. This is because for extremely low levels of noise (0.01% in the isotropic case, 0.1% anisotropic case) the solution space becomes flat (i.e., no real preference for any solution), and the optimal choice of $$$f$$$ tends towards $$$f=0$$$.

Figures 3-4 demonstrate using a multi-shell scheme that for low levels of noise (< 1%) the WLLS estimated solution is well defined with a marked dip in the residuals around the ground truth of $$$f=\frac{1}{2}$$$. However, as the noise-level grows larger we again observe that the curves become more elongated and flat (although equally sensitive) in the isotropic and anisotropic case. Note that the noise levels for which this behaviour manifested was 10-100 times greater than with the single-shell scheme.

Direct fitting methods such as the WLLS assume well-posed solution space, where the minimal error term corresponds with the optimal solution. Our results show that the free-water model is well posed only for high quality data with low noise levels. As the noise levels worsen the solution space flattens and the optimal solution is biased towards smaller $$$f$$$. Consequently, the solution increasingly neglects the free-water component. This is likely due to the tensor in the tissue compartment having more degrees of freedom and thus can better (over-) fit the data when the noise begins to dominate the signal. In addition we note that the fit becomes unstable faster for the single-shell isotropic case compared to the anisotropic case.

Our results suggest that direct fitting of multi-shell data offer 1-2 orders of magnitude better noise sensitivity over single-shell. However, direct fitting becomes unstable in noise levels that may be encountered in clinical and research settings. In future work we wish to investigate how multi-shell results can be improved further by using regularization [3] or statistical priors [4].