Fast and Accurate Initialization of the Free-Water Imaging Model Parameters from Multi-Shell Diffusion MRI

Abstract

Cerebrospinal fluid partial volume effect is a known bias in the estimation of Diffusion Tensor Imaging (DTI) parameters from diffusion MRI data. The Free-Water Imaging model for diffusion MRI data adds a second compartment to the DTI model, which explicitly accounts for the signal contribution of extracellular free-water, such as cerebrospinal fluid. As a result the DTI parameters obtained through the free-water model are corrected for partial volume effects, and thus better represent tissue microstructure. In addition, the model estimates the fractional volume of free-water, and can be used to monitor changes in the extracellular space. Under certain assumptions, the model can be estimated from single-shell diffusion MRI data. However, by using data from multi-shell diffusion acquisitions, these assumptions can be relaxed, and the fit becomes more robust. Nevertheless, fitting the model to multi-shell data requires high computational cost, with a non-linear iterative minimization, which has to be initialized close enough to the global minimum to avoid local minima and to robustly estimate the model parameters. Here we investigate the properties of the main initialization approaches that are currently being used, and suggest new fast approaches to improve the initial estimates of the model parameters. We show that our proposed approaches provide a fast and accurate initial approximation of the model parameters, which is very close to the final solution. We demonstrate that the proposed initializations improve the final outcome of non-linear model fitting.

1 |. NTRODUCTION

Diffusion MRI is a non-invasive imaging methodology sensitive to the micron scale displacement of water molecules. The size of the displacement and its directional dependency reflect hindrance or restriction to the water molecules motion, yielding unique information, especially for white-matter brain structures¹, about tissue microstructural properties and the organization of white-matter bundles². Currently, Diffusion Tensor Imaging (DTI)^3,4 is the most popular way to infer microstructural properties from diffusion MRI. In DTI, the three-dimensional displacement is described by a single tensor compartment. However, the voxel resolution in diffusion MRI (typically in the mm scale) is coarse relative to tissue microstructure of interest in the brain (typically in micron scale). This leads to partial volume effects, i.e., misinterpretation of microstructural properties inferred by DTI, when multiple tissue types are present in the same voxel⁵.

The Free-Water Imaging model⁶ is an extension of the DTI model that describes the diffusion MRI signal as a weighted mixture of a tensor compartment representing brain tissue, and a second compartment modeled by an isotropic tensor with diffusivity fixed to that of free-water at body temperature, i.e., 3_.10⁻³mm²_/s⁵. Free-water is defined as water molecules that do not experience hindrance or restriction during the diffusion experiment. For typical diffusion times (in the order of 50ms), free-water in the brain can only be found in the extracellular space, i.e., plasma, cerebrospinal fluid in the ventricles and around the parenchyma of the brain, and also, potentially, in the interstitial space, e.g., when vasogenic edema occurs, or if there are large enough pockets that allow free diffusion⁷. Similar to the two compartment model, other diffusion models (e.g., diffusion basis spectrum imaging⁸, neurite orientation dispersion and density imaging^9,10, and multiple fascicle models¹¹) also estimate an explicit free-water compartment. The relative volume of the free-water compartment could provide important information about pathological processes that modify the extracellular space; such as edema⁶, neuroinflammation⁸ and atrophy¹² that appear in e.g., aging¹², schizophrenia¹³, Multiple Sclerosis⁸ and Alzheimer’s disease¹⁴. Eliminating the contribution of the free-water compartment provides DTI measures that are corrected for the partial volume effect of the free-water. The corrected measures improve the specificity of the DTI metrics to brain tissue¹³ and help reduce reproducibility errors¹⁵.

The two compartment Free-Water Imaging model adds a single additional parameter to the DTI model, but nevertheless, the estimation of the model is ill-posed for a single-shell acquisition data, requiring a regularized fit approach, which applies additional constraints when estimating the model parameters⁶. For multiple b-values, the estimation remains difficult^16,17,18 and requires non-linear minimization techniques with high computational burden. Although powerful, non-linear minimization techniques typically suffer from a number of significant drawbacks including (a) difficulties converging to a global minimum among many local minima; (b) high computational expense; and (c) difficulties constraining the solutions to be physically plausible. Providing an initial guess that is close to the global minimum helps the non-linear solvers avoid converging to local minima and may also improve the speed of convergence. There are currently two proposed initialization approaches for fitting the free-water model using multi-shell data. One approach reduces the free-water model to a large number of linear DTI estimations, and then uses brute force to select the best solution¹⁸ as an initialization for a non-linear minimization. The second approach uses the high b-values, where signal from free-water disappears, to estimate the free-water corrected DTI values. This approach then uses the low b-values to estimate the contribution of free-water¹⁷. The estimated tensor and free-water contribution are then used to initialize a non-linear minimization.

In this work, we analyze the properties of the two previously proposed fast initialization approaches in noisy experiments and propose improvements that provide more accurate fast estimations, which may reduce the need for explicit non-linear solvers.

2 |. THEORY

The single tensor DTI model³,

$${\text{s}}_{\text{i}}={\text{s}}_0\exp (-{\text{b}}_{\text{i}}{\mathbf{g}}_{\mathbf{\text{i}}}^{\top }\mathbf{D}{\mathbf{g}}_{\mathbf{\text{i}}}),$$ (1)

describes the relationship between the voxelwise 3 × 3 symmetric diffusion tensor D, the n diffusion gradient directions ${\mathbf{g}}_{\mathbf{\text{i}}}={[{\text{g}}_{\text{ix}}{\text{,g}}_{\text{iy}}{\text{,g}}_{\text{iz}}]}^{\top }$, and the diffusion weighted image s_i (in this paper, upper case bold font is used to indicate matrices and lower case bold font indicates column vectors). We denote the voxelwise signal of the non-diffusion weighted baseline image as s₀, and b_i is the b-value of each diffusion weighted image (unlike b = 0 of the non diffusion weighted images), which may be constant for the whole scan (single-shell), or may vary between diffusion weighted images (multi-shell). The n measurements form a set of n equations, which can be linearized by defining a design matrix

$$\mathbf{\text{R}}=\left[\begin{array}{ccccccc}1& -{\text{b}}_1{\text{g}}_{1\text{x}}^2& -2{\text{b}}_1{\text{g}}_{1\text{x}}{\text{g}}_{1\text{y}}& -2{\text{b}}_1{\text{g}}_{1\text{x}}{\text{g}}_{1\text{z}}& -{\text{b}}_1{\text{g}}_{1\text{y}}^2& -2{\text{b}}_1{\text{g}}_{1\text{y}}{\text{g}}_{1\text{z}}& -{\text{b}}_1{\text{g}}_{1\text{z}}^2\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& -{\text{b}}_{\text{n}}{\text{g}}_{\text{nx}}^2& -2{\text{b}}_{\text{n}}{\text{g}}_{\text{nx}}{\text{g}}_{\text{ny}}& -2{\text{b}}_{\text{n}}{\text{g}}_{\text{nx}}{\text{g}}_{\text{nz}}& -{\text{b}}_{\text{n}}{\text{g}}_{\text{ny}}^2& -2{\text{b}}_{\text{n}}{\text{g}}_{\text{ny}}{\text{g}}_{\text{nz}}& -{\text{b}}_{\text{n}}{\text{g}}_{\text{n}2}^2\end{array}\right]$$

yielding for each voxel the equivalent linear system of equations

$$\mathbf{\text{Rx}}=\log (\mathbf{\text{s}})$$

with the unknown $\mathbf{x}={[\log ({\text{s}}_0){\text{D}}_{\text{xx}}{\text{D}}_{\text{xy}}{\text{D}}_{\text{xz}}{\text{D}}_{\text{yy}}{\text{D}}_{\text{yz}}{\text{D}}_{\text{zz}}]}^{\top }$ This linear system can be solved given a measured diffusion signal s as a standard weighted linear least-squares (LLS) problem by defining the residual

$${ϵ}_{\text{lls}}=\Vert \mathbf{W}(\mathbf{R}\mathbf{x}-\log (\mathbf{s})){\Vert }_2.$$ (2)

and finding x that minimizes the square of ϵ_lls. The diagonal matrix W = diag(w_i) is a weighting matrix that could be the identity matrix, i.e., w_i = 1, or have entries that set a weight for each diffusion image. For example ${\text{w}}_{\text{i}}={\text{s}}_{\text{i}}^2$ was proposed to correct the heteroscedasticity introduced by the logarithmic transform.^3,19,18

The solution can be obtained using, for example, the Normal Equation (and the pseudo-inverse), the QR algorithm, or the singular value decomposition for the unknown tensor D and baseline s₀. Using the pseudo-inverse, the solution has a closed form expression

$$\mathbf{x}={({\mathbf{R}}^{\top }\mathbf{W}\mathbf{R})}^{-1}{\mathbf{R}}^{\top }\mathbf{W}\log (\mathbf{s}).$$ (3)

2.1 |. The Free-Water Imaging model

The Free-Water Imaging model augments the DTI model by adding a compartment describing a fast diffusing free-water element⁶

$${\text{s}}_{\text{i}}{=\text{s}}_{\text{0}}[(1-\text{f})\text{exp}({-\text{b}}_{\text{i}}{\mathbf{\text{g}}}_{\mathbf{\text{i}}}^{\top }\mathbf{D}{\mathbf{g}}_{\mathbf{\text{i}}})+\text{f exp}\left({-\text{b}}_{\text{i}}{\text{D}}_{\text{fw}}\right)].$$ (4)

The free-water compartment is modeled using a constant apparent diffusion coefficient, D_fw = 3 · 10⁻³mm²/s, equal to the expected diffusivity of water at body temperature. The remaining tissue compartment, as in DTI, is modeled using a diffusion tensor. The Free-Water Imaging model adds one new parameter to the DTI model, f ∈ [0, 1], which is the volume fraction of the free-water compartment, setting the weighted contribution of the two model compartments.

Even though the free-water imaging model adds only one more free parameter to the DTI model, the fit becomes ill-posed^18,20. A robust fit of the model requires non-linear minimization of the residual

$${ϵ}_{\text{nls}}=\sqrt{\sum _{\text{i}}{({\text{s}}_{\text{i}}-{\text{s}}_0[(1-\text{f})\exp (-{\text{b}}_{\text{i}}{\mathbf{g}}_{\mathbf{\text{i}}}^{\top }\mathbf{D}{\mathbf{g}}_{\mathbf{\text{i}}})+\text{f}\exp (-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})])}^2}.$$ (5)

Previous studies applied Newton based approaches¹⁹, e.g., the Levenberg-Marquardt method¹⁸, or derivative free-methods, such as BOBYQA¹¹, as non-linear minimizers. Faster and less accurate approaches to fit the model have been proposed as initial guesses for the non-linear process in order to direct the non-linear fit closer to the desired solution^17,18. In the sections below, we investigate the analytic properties of these initialization approaches, as well as suggest improved initialization approaches.

2.2 |. Linearization of the Free-Water Imaging model

The Free-Water Imaging model can be deconstructed into two separate linear problems: estimating f for a given D, and estimating D for a given f ^17,18,20. We observe that Eq. (4) can be re-arranged into

$$\frac{{\text{s}}_{\text{i}}-{\text{s}}_0\text{f exp}(-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})}{1-\text{f}}={\text{s}}_{\text{0}}\exp (-{\text{b}}_{\text{i}}{\mathbf{g}}_{\mathbf{\text{i}}}^{\top }\mathbf{D}{\mathbf{g}}_{\mathbf{\text{i}}}).$$ (6)

When 0 < f < 1 is given, the left side of Eq. (6) is the diffusion weighted signal expected to be measured from the tissue compartment alone. We note that when f = 0 the model is identical to the DTI model in Eq. (1). Defining the left hand side of Eq. (6) as a new variable ${\widehat{\text{s}}}_{\text{i}}$, i.e.,:

$${\widehat{\text{s}}}_{\text{i}}:=\frac{{\text{s}}_{\text{i}}-{\text{s}}_0\text{f}\exp (-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})}{1-\text{f}},$$ (7)

we get

$${\widehat{\text{s}}}_{\text{i}}={\text{s}}_0\exp (-{\text{b}}_{\text{i}}{\mathbf{g}}_{\mathbf{\text{i}}}^{\top }\mathbf{D}{\mathbf{g}}_{\mathbf{\text{i}}}),$$

which has the same form as Eq. (1). Therefore, if f is given, D and s₀ can be linearly estimated by first calculating $\widehat{\mathbf{\text{s}}}$ from s (using Eq. (7)), and then replacing s with $\widehat{\mathbf{\text{s}}}$ in Eq. (3), which provides a LLS solution, as in the DTI case. Here, the quantity $\widehat{\mathbf{\text{s}}}$ can be interpreted as the remaining signal, after the contribution of free-water has been removed.

Algorithm 1.

The voxelwise search algorithm to estimate a solution $(\widehat{\text{f}},\widehat{\text{D}})$ for the free-water diffusion model

Require: Diffusion weighted images si for a given voxel
1:	procedure SearchF(si)
2:	$\widehat{ϵ}\leftarrow \infty$
3:	for all f ∈ [0, 1) do
4:	${\widehat{\text{s}}}_{\text{i}}\leftarrow \frac{{\text{s}}_{\text{i}}-{\text{s}}_0\text{f}\exp (-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})}{1-\text{f}}$
5:	D ← fit a tensor from $\widehat{\mathbf{\text{s}}}$ (instead of s), using Eq. (3).
6:	$ϵ$ ← the goodness-of-fit measure of (f, D)
7:	if $ϵ<\widehat{ϵ}$ & Positive(D) then
8:	$\widehat{ϵ}\leftarrow ϵ$
9:	$\widehat{\text{f}}\leftarrow \text{f}$
10:	$\widehat{\mathbf{D}}\leftarrow \mathbf{D}$
11:	end if
12:	end for
13:	return $(\widehat{\text{F}},\widehat{\mathbf{\text{D}}})$
14:	end procedure
15:	procedure Positive(D)
16:	return ${\lambda }_3(\mathbf{D})\ge 0$
17:	end procedure

On the other hand, if the diffusion tensor D is given, then the unknown scalar f can be calculated by rearranging Eq. (4) into

$$\begin{gathered} \left[\exp (-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})-\exp (-{\text{b}}_{\text{i}}{\mathbf{g}}_{\mathbf{\text{i}}}^{\top }\mathbf{D}{\mathbf{g}}_{\mathbf{\text{i}}})\right]\text{f}=\left[\frac{{\text{s}}_{\text{i}}}{{\text{s}}_0}-\exp (-{\text{b}}_{\text{i}}{\mathbf{g}}_{\mathbf{\text{i}}}^{\top }\mathbf{D}{\mathbf{g}}_{\mathbf{\text{i}}})\right]. \\ {\text{p}}_{\text{i}}\text{f}={\text{q}}_{\text{i}} \end{gathered}$$ (8)

This linear system of equations (one equation for each diffusion weighted image) can be solved as a simple linear regression problem, which has a closed-form expression for its solution

$$\text{f}=\frac{{\mathbf{p}}^{\top }\mathbf{q}}{{\mathbf{p}}^{\top }\mathbf{p}}.$$

2.3 |. The LLS search initialization

The linearization approach was utilized by Hoy et. al.¹⁸ to form a search algorithm that samples a grid of f values over the entire range of 0 ≤ f < 1. The procedure (outlined in Alg. (1)) finds a tensor D for each f by calculating $\widehat{\mathbf{\text{s}}}$ (Eq. (7)), and then replacing s with $\widehat{\mathbf{\text{s}}}$ in Eq. (3). The solution $(\widehat{\text{f}},\widehat{\mathbf{D}})$, is selected as the one with the smallest residual ${ϵ}_{\text{lls}}$ of all pairs tested. This solution can then be used as an initialization for a non-linear procedure. The additional non-linear step is required since the search algorithm is limited to the preselected grid of f values, whereas the optimal result likely lies in between the sampled grid points. The non-linear step is also useful to account for the heteroscedasticity stemming from the logarithmic transform involved in the linearization step.

2.3.1 |. Noise propagation of the LLS approach

While the above-mentioned LLS approach was previously used as the basis for the LLS search approach, the noise propagation properties of this approach were not yet studied. Therefore, in this sub-section we provide a noise propagation analysis for the LLS approach. We consider (s + η), i.e., a diffusion signal s perturbed by noise η. By substituting s with (s + η) in the left hand side of Eq. (6), we define $\tilde{\mathbf{s}}$, the noisy signal of the tissue compartment as:

$$\begin{gathered} {\tilde{\text{s}}}_{\text{i}}:=\frac{({\text{s}}_{\text{i}}+{\eta }_{\text{i}})-{\text{s}}_0\text{f}\exp (-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})}{1-\text{f}}=\frac{{\text{s}}_{\text{i}}-{\text{s}}_0\text{f}\exp (-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})}{1-\text{f}}+\frac{{\eta }_{\text{i}}}{1-\text{f}} \\ ={\widehat{\text{s}}}_{\text{i}}+\frac{{\eta }_{\text{i}}}{1-\text{f}}. \end{gathered}$$

Again, by taking the logarithm of $\tilde{\mathbf{s}}$, we can solve for D and s₀ using Eq. (2). Using the identity $\log (\text{a}+\text{b})=\log (\text{a})+\log (1+\frac{\text{b}}{\text{a}})$, we see that

$$\begin{gathered} \log ({\tilde{\text{s}}}_i)=\log \left({\widehat{\text{s}}}_i+\frac{{\eta }_i}{1-\text{f}}\right)=\log ({\widehat{\text{s}}}_i)+\log \left(1+\frac{{\eta }_i}{{\widehat{\text{s}}}_i(1-\text{f})}\right) \\ =\log ({\widehat{\text{s}}}_{\text{i}})+\log \left(1+\frac{{\eta }_{\text{i}}}{{\text{s}}_{\text{i}}-{\text{s}}_0\text{f}\exp (-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})}\right), \end{gathered}$$

which means that for the Free-Water Imaging model, the LLS procedure minimizes the residual

$${ϵ}_{\text{lls}}={\Vert \mathbf{W}\left(\mathbf{\text{Rx}}-\log (\widehat{\mathbf{s}})-\log \left(1+\frac{{\eta }_{\text{i}}}{{\text{s}}_{\text{i}}-{\text{s}}_0\text{f}\exp (-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})}\right)\right)\Vert }_2.$$

This formulation reveals a key insight into the LLS minimization of the Free-Water Imaging model. First, we observe that the term $\mathbf{\text{Rx}}-\log (\widehat{\mathbf{\text{s}}})$ is minimized for the ground truth solution $(\mathbf{D},\text{f})=({\mathbf{D}}^{*},{\text{f}}^{*})$, yielding $\mathbf{R}\mathbf{x}-\log (\widehat{\mathbf{s}})=0$. Consequently, the residual ${ϵ}_{\text{lls}}$ will be exactly equal to the norm of the weighted noise term. However, this noise term is also coupled with f. We see that as f is decreased, the denominator grows larger effectively reducing the noise term. The noise term will thus be minimal and equal to w_i $\log \left(1+\frac{{\eta }_{\text{i}}}{{\text{s}}_{\text{i}}}\right)$ when f = 0. The separation of terms implies that there are two forces in the minimization of the LLS approach for the free-water imaging model: one that pulls towards the ground truth f = f* and another one that pulls towards f = 0.

To summarize, the noise propagation analysis of the LLS approach predicts that the solution f will represent a trade-off between the true solution (f = f*) and a solution that neglects the free-water component (f = 0). It further predicts that as the noise level η_i grows larger, the preference for f = 0 will dominate.

2.3.2 |. Proposed improvement of the LLS search approach: The NLS search approach

The LLS search approach determines the best solution by a scalar goodness of fit criteria. In the LLS approach above, the criteria was the residual ϵ_lls from Eq. (2), which explicitly minimizes the linear model. However, given our LLS noise propagation analysis above, as noise grows the ϵ_lls measure could be biased towards under-estimating the free-water component.

We therefore suggest the NLS search approach, i.e., using ϵ_nls from Eq. (5) instead of ϵ_lls as the goodness of fit measure. Similar to the noise propagation analysis in Section (2.3.1) above, we can examine the effect of noise when using the criteria defined in Eq. (5) by considering a noisy signal $\widetilde{\mathbf{\text{s}}}=\mathbf{\text{s}}+\mathit{\eta }$, giving the minimization problem

$${ϵ}_{\text{nls}}=\sqrt{\sum _{\text{i}}{({\text{s}}_{\text{i}}+{\eta }_{\text{i}}-{\text{s}}_0(1-\text{f})\exp (-{\text{b}}_{\text{i}}{\mathbf{\text{g}}}_{\mathbf{\text{i}}}^{\top }{\mathbf{\text{Dg}}}_{\mathbf{\text{i}}})+\text{f}\exp (-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}}))}^2}$$

We observe that if (f, D) = (f*, D*), then ${ϵ}_{\text{nls}}=\Vert \mathit{\eta }{\Vert }_2$ which means that the effect of the noise is no longer a function of f, avoiding the bias seen with ϵ_lls.

It is important to note that changing the goodness of fit criteria does not alter the calculation of each individual tensor D from a given f, which is still done by linear minimization of ϵ_lls. However, by altering the goodness of fit criteria, the NLS search approach now selects the solution, $(\widehat{\text{f}},\widehat{\mathbf{\text{D}}})$, that minimizes the non-linear cost function ϵ_nls (instead of the linear cost function ϵ_lls) from the same collection of linearly estimated candidates.

2.3.3 |. Ensuring positive semi-definite diffusion tensors

To maintain physically plausible tensors, the fitted diffusion tensor should be (a) symmetric and (b) positive semi-definite (all eigenvalues are nonnegative). Constraint (a) is guaranteed since 6 unique elements are estimated instead of the complete 3 × 3 diffusion tensor D. In the context of Alg. (1), we propose to enforce the positivity constraint by disregarding any possible solution which has a tensor with a negative eigenvalue. This approach, implemented in Alg. (1), guarantees a positive semi-definite solution as long as a positive semi-definite candidate exists. In the rare occasion where all candidates are negative, the voxel may be omitted from consideration.

2.4 |. The High-Low initialization approach

The previously proposed High-Low initialization¹⁷ outlined in the procedure HiLow in Alg. (2), relies on the observation that the free-water term contributes proportionally less at higher b_i-value shells¹⁷. Therefore HiLow uses only the high b_i-values (requires at least two different high b-values) to estimate D. This corresponds to setting ${\text{w}}_{\text{i}}=0{\text{if b}}_{\text{i}}\le {\text{T}}_{\text{high}}$, and then applying Eq. (3) to linearly solve for D and s₀. The selection of T_high depends on the available b_i-value shells, with the general notion that it should be high enough to suppress most of the signal originating from free-water relative to the signal originating from the tissue; e.g., b_i = 600s/mm² suppresses more than 80% of the free-water signal compared to less than 40% of the tissue signal and is thus sufficiently high, although higher T_high is better. All images with b < T_high, including any acquired b = 0 images, are not included in this calculation. DWIs with b > 2000s/mm² should be excluded to prevent from non-Gaussian effects. Once D is computed, the low b-value measurements $\{{\text{s}}_{\text{j}}{:\text{b}}_{\text{j}}\le {\text{T}}_{\text{low}}\}$ are used with Eq. (8) to estimate f. The T_low threshold chooses the b-value range where the free-water has a significant contribution e.g., ${\text{T}}_{\text{low}}\le 800\text{s}/{\text{mm}}^2$ where more than 90% of the free-water signal is decayed, while less than 50% of the tissue signal is decayed.

The advantages of the High-Low approach over the search algorithm approach are (a) speed, because the DTI sub-problem is solved only once rather than a large number of times, and (b) explicitly taking advantage of the different information embedded in different b_i-shells. However, an important limitation of this approach is that although the higher b_i-value shells are less affected by the free-water term, signal from free-water still has some contribution, and more so when T_high is lower. Therefore, the estimated D would still have some free-water contribution, effectively

Algorithm 2.

Using low and high b-value shells to estimate a solution (f_k, D_k)

Require: Diffusion weighted images ${\text{s}}_{\text{i}}|0<\text{i}<\text{n}$, for a given voxel
1:	procedure HiLow(s)
2:	$\mathcal{H}\leftarrow \{\text{i}:{\text{b}}_{\text{i}}>{\text{T}}_{\text{high}}\}$
3:	$\left({\mathbf{\text{D}}}_0,{\text{s}}_0\right)$ fit the tensor model (Eq. (3)) using ${\text{w}}_{\text{i}}=0\text{for i}\notin \mathcal{H}$
4:	$\mathcal{L}\leftarrow \{\text{i}:{\text{b}}_{\text{i}}\le {\text{T}}_{\text{low}}\}$
5:	${\text{f}}_{\text{0}}\leftarrow$ estimate the volume fraction (Eq. (8)) from s using only $\text{i}\in \mathcal{L},\text{and}{\mathbf{\text{D}}}_{\text{0}}$
6:	return (f0, D0, s0)
7:	end procedure
8:	procedure HiLowDownhill(s)
9:	(f0, D0, s0) ← HiLow(s)
10:	${\mathbf{\text{D}}}_0^{+}\leftarrow$ StayPositive(10−3 · I, D0)
11:	for $k\leftarrow \{1,2,\dots ,\text{m}\}\mathbf{\text{do}}$
12:	${\widehat{\text{s}}}_{\text{i}}\leftarrow \frac{{\text{s}}_{\text{i}}-{\text{s}}_0{\text{f}}_{\text{k}-1}\text{exp}(-{\text{b}}_{\text{i}}{\text{D}}_{\text{fw}})}{1-{\text{f}}_{\text{k}-1}}$
13:	${\mathbf{\text{D}}}_{\text{k}}$ ← fit the tensor model (Eq. (3)) from $\widehat{\mathbf{s}}$
14:	${\mathbf{\text{D}}}_{\text{k}}^{+}\leftarrow$ StayPositive $({\mathbf{\text{D}}}_{\text{k}-1}^{+},{\mathbf{\text{D}}}_{\text{k}})$
15:	${\mathrm{f}}_{\text{k}}\leftarrow$ estimate the volume fraction (Eq. (8)) from s using ${\mathbf{\text{D}}}_{\text{k}}^{+}$
16:	if ${ϵ}_{\text{nls}}({\text{f}}_{\text{k}}{,\mathbf{\text{D}}}_{\text{k}}^{\text{+}})>{ϵ}_{\text{nls}}({\text{f}}_{\text{k}-1}{,\mathbf{\text{D}}}_{\text{k}-1}^{+})\mathbf{\text{then}}$
17:	return $({\text{f}}_{\text{k}-1},{\mathbf{\text{D}}}_{\text{k}-1}^{+})$
18:	end if
19:	end for
20:	return $({\text{f}}_{\text{k}}{,\mathbf{\text{D}}}_{\text{k}}^{\text{+}})$
21:	end procedure
22:	procedure StayPositlve(Dk − 1, Dk)
23:	if Positive(Dk) then
24:	return Dk
25:	end if
26:	if Distance(Dk − 1, Dk) < ϵ then
27:	return Dk − 1
28:	else
29:	$\mathbf{\text{M}}\leftarrow \frac{1}{2}\left({\mathbf{\text{D}}}_{\text{k}-1}+{\mathbf{\text{D}}}_{\text{k}}\right)$
30:	if Positive(M) then
31:	return StayPositive(M, Dk)
32:	else
33:	return StayPositlve(Dk − 1, M)
34:	end if
35:	end if
36:	end procedure

overestimating diffusivity and underestimating f. Another limitation is that the estimation of f and D uses subsets of the DWIs (based on high or low b_i value shells), effectively lowering the number of data points, n, for each step and thus making each estimate more sensitive to noise. Finally, in the original approach there were no means to explicitly ensure that the estimated diffusion tensor D has non-negative eigenvalues.

2.4.1 |. Proposed modification to the High-Low approach: High-Low Downhill

In this subsection we propose an algorithm that improves the High-Low approach by repeatedly re-estimating D and f so that the final estimates provide better fit via the goodness of fit criteria in Eq. (5). Our proposed HiLowDownhill method can be considered as a simplification of the more general Variable Projection non-linear optimization procedure ^21,22 which among other uses was previously proposed to estimate the intravoxel incoherent motion ²³ model and a crossing fibers model ²⁴. In our application the procedure is simplified significantly by having two linear partial models (solving for f, and solving for D).

Our algorithm, summarized in the HiLowDownhill procedure in Alg. (2), starts from the initial values (f_k−1,D_k−1) = (f₀, D₀) computed by the HiLow procedure. Then, we again deconstruct the non-linear problem to the two linear sub-problems. By fixing f_{k − 1}, we apply LLS via Eq. (2) and Eq. (3), using all the DWIs to calculate a new tensor estimate D_k. The obtained D_k is expected to provide the minimal residual (in a least-squares sense) for the fixed f_{k − 1}. Therefore, the residual of (f_{k − 1}, D_k) is less than or equal to the residual of (f_{k − 1}, D_{k − 1}). If the residual is lower, we continue by fixing the obtained D_k, and then linearly calculate f_k using Eq. (8). Again, the estimated f_k is expected to provide the minimal residual for the fixed D_k and thus the residual of (f_k, D_k) is less than or equal to the residual of (f_{k − 1}, D_k). We continue with this iterative process until a maximum number of iterations, m, is reached, or if the residual no longer reduces. Therefore, this design assures that as long as D₀ is a positive semi-definite tensor, the residual of HiLowDownhill improves through the iterative process and that it will always be less than or equal to the residual of HiLow.

2.4.2 |. Ensuring positive semi-definite diffusion tensors

To ensure that the tensors calculated by the HiLowDownhill have non-negative eigenvalues, we propose a line-search algorithm that replaces any negative tensor with a non-negative tensor. The approach, outlined in the procedure StayPositive of Alg. (2) assumes that D_{k − 1} is positive semi-definite. It then first checks if D_k is positive by checking if the smallest eigenvalue ${\lambda }_3({\mathbf{D}}_{\text{k}})\ge 0$. If so, D_k is kept. Otherwise, we use a binary search to find a positive semi-definite tensor which is closest to D_{k − 1}, and which lies on a line segment from D_{k − 1} to D_k (based on the Euclidean distance²⁵, i.e., Distance $({\mathbf{\text{D}}}_{\text{a}},{\mathbf{\text{D}}}_{\text{b}})={\Vert {\mathbf{\text{D}}}_{\text{a}}-{\mathbf{\text{D}}}_{\text{b}}\Vert }_{\text{F}}$, where $\Vert \cdot {\Vert }_{\text{F}}$ denotes the Frobenius norm). The barycentric search problem is parameterized by defining

$${\widehat{\mathbf{D}}}_k^{\alpha }=(1-\alpha ){\mathbf{D}}_{k-1}+\alpha {\mathbf{D}}_k,$$

where α ∈ [0, 1]. Then, the solution is the largest α = α_max such that ${\widehat{\mathbf{D}}}_{\text{k}}^{\alpha }$ is positive semi-definite. The procedure StayPositive of Alg. (2) outlines a simple implementation for the binary search, which stops once converging to a poisitive tensor (i.e., change in D across iterations is sufficiently small; in our implementation we use $ϵ$ = 10⁻¹⁴). Having found the desired solution ${\widehat{\mathbf{D}}}_{\text{k}}^{{\alpha }_{\max }}$, this tensor is used as a positive semi-definite approximation to D_k.

This procedure guarantees to output a positive semi-definite tensor instead of D_K, as long as the input D_K−1 is positive semi-definite. This requires that the initial estimation, D₀ is also positive. Voxels with negative D₀ may be eliminated from analysis. Alternatively, a negative D₀ can be replaced by applying the StayPositive-procedure on D₀ along with any selected positive tensor. In our implementation, if D₀ is negative, we replace it with ${\mathbf{\text{D}}}_0^{+}\leftarrow$ StayPositive(10⁻³ · I, D₀), where I is the identity matrix.

3 |. METHODS

The four different initialization approaches discussed are compared using a number of experiments: a single voxel simulation, a synthetic grid simulation, a simulated brain, and an in-vivo brain. The same diffusion gradient table was used for all experiments. The table represented a multi-shell acquisition scheme with 65 gradient directions designed as nested platonic solids so that (a) each shell is rotationally invariant and (b) all shells complement each other to form a rotationally invariant scheme^17,26. The 65 b-values were ${\text{b}}_{\text{i}}\in \{3\times 50,6\times 200,10\times 500,30\times 900,16\times 1400\}{\text{mm}}^2$/s. A single b = 0 image was additionally acquired. 6,10 and 30 directions are obtained from the non-colinear vertices of an icosahedron, dodecahedron, and truncated icosahedron respectively. 16 directions are obtained by combining the 6 and 10 directions, which can be combined since icosahedron and dodecahedron have interchanged faces and vertices (the center of a face in the icosahedron is a vertex in the dodecahedron)²⁶. The three directions for the lowest shell were selected as an orthogonal basis. This gradient scheme was designed for quantitative analyses, rather than for tractography studies. For tractography studies, higher orientation resolution in the highest b-shell is required.

3.1 |. Implementation

To compare the different results, we applied the four initialization algorithms described above to estimate D and f from the different datasets. All algorithms were implemented and run in Matlab version 9.1.0 on Mac OS X Sierra and were not optimized for speed. The algorithms were: (a) SearchF_lls, implementing Alg. (1) with ${ϵ}_{\text{lls}}$ as a goodness of fit criteria, (b) SearchF_nls, implementing Alg. (1) with ${ϵ}_{\text{lns}}$ as a goodness of fit criteria, (c) HiLow, and (d) HiLowDownhill, implemented as outlined in Alg. (2).

In addition to the comparison of the four different estimation algorithms, the results obtained from each algorithm were used to initialize a non-linear minimization solver. For this, the non-linear least square fitting procedure of the Free-Water Imaging model provided by the Diffusion Imaging in Python package (dipy.org) was used²⁰. This reference algorithm is based on a modified Levenberg-Marquardt algorithm²⁷, using explicit calculation of the model Jacobian²⁰.

For all algorithms, voxels with mean diffusivity equal to or larger than that of free-water are automatically assigned with f = 1 and D = 0 and are omitted from further fit.

3.2 |. Data simulation

Given a ground truth free-water fraction f* and diffusion tensor D*, we calculated the ground truth DWIs, s_i, using Eq. (4). Monte Carlo simulations were performed by adding noise to generate Rician distributed ${\tilde{s}}_i=\sqrt{{\left(s_i+{\nu }_1\right)}^2+{\nu }_2^2}$ where ν₁ and ν₂ were picked from a normal distribution $\mathcal{N}(0,\text{p}\cdot {\text{s}}_0)$ with zero mean and standard deviation expressed as a fraction $0\le \text{p}\ll 1$ of the baseline s₀. We calculated the nominal signal-to-noise ratio (SNR) as $\text{SNR}=\frac{1}{\text{p}}$.

3.2.1 |. Single voxel dataset

The simulated single voxel dataset had f* = 0.4, and the tensor D* was a diagonal matrix with eigenvalues $\lambda =\{1.6,0.5,0.3\}\cdot {10}^{-3}{\text{mm}}^2/\text{s}$. These eigenvalues were chosen to replicate the simulations reported by Hoy et. Al.¹⁸. Thirteen different noise levels ($0\%,0.5\%,1\%,\dots ,6\%$ of the base-line, corresponding to SNRs of $\{\infty ,200,100,67,50,\cdots ,17\}$) were simulated. For each noise level, we performed 100,000 repeated simulations. Each simulation resulted with a pair (f, D). We then report the mean and standard deviation of f across all simulations.

For all SearchF_lls and SearchF_nls experiments, D was calculated for 100 equidistant f ∈ [0, 1); for each pair (f, D) we calculated the residuals ϵ_lls and ϵ_nls, respectively, and identified the pair with the minimal residual. For the HiLow and HiLowDownhill, we estimated (f, D) using T_high = T_low = 800, i.e., the highest b-value in the low-b group was 500s/mm², and the lowest b-value in the high-b group was 900s/mm². The same T_high and T_low values are used in all experiments below.

3.2.2 |. The Synthetic Grid

To compare the four initialization approaches over a range of D and f combinations, we created a synthetic phantom comprising a volume of 50 × 50 × 50 voxels with varying orientation, anisotropy, and f values between 0 and 1 (See Appendix A for more details). Following¹⁸, all tensors had a trace of 2.4 10⁻³mm²/s. This synthetic grid yielded a uniform distribution of tensors and f values.

The noise levels simulated were 0%, 0.5%m,···,20%, and comparisons between the algorithms were made for (a) ϵ_nls residuals, (b) the difference between the estimated free-water fraction and its ground truth, and (c) the difference between the estimated tensor FA and the FA of the ground truth.

3.2.3 |. Simulated and in vivo Brain Datasets

To obtain a more realistic distribution of f and D values while maintaining a known ground truth, we simulated a brain dataset. To do so, we collected DWI data from a healthy human volunteer on a Siemens 1.5T scanner. The data was collected with 2.5mm isotropic voxels, 128 × 128 matrix and 56 slices. TE/TR were 85/8500ms. The estimated baseline SNR was 34. The protocol included the same 65 gradient table as described above and took 9:40 minutes to complete. The protocol and acquisition were approved by the local IRB committee, and the subject signed a consent form. The image was manually masked to include the brain. Following masking, the in vivo brain dataset included 104296 voxels. Motion and eddy current correction were performed using non-linear registration of each DWI image to the b = 0 image.

To estimate reference f and D solutions, we used the previously proposed non-linear and regularized approach^6,17. The obtained f and D values were then used as references to generate ground truth DWIs, to which noise was added. For the experiments below we consider two noise levels, one corresponding to a low noise level scan with SNR 33 (3% noise), and another with medium noise level of SNR 20 (5% noise).

As a final evaluation, we applied the initialization algorithms directly on the pre-processed scanned DWIs. It is important to note that in this experiment there is not a ground truth solution for comparison. Evaluation is thus limited to visual inspection and comparison of model residuals $({ϵ}_{\text{nls}})$, i.e., the difference between the signal predicted by the fitted model parameters and the actual acquired signal.

4 |. RESULTS

4.1 |. The Single Voxel Dataset and Noise Sensitivity

To demonstrate the different initialization approaches under varying noise levels, we first consider a set of experiments using a single voxel representing white matter. Fig. (1) shows the estimated f for the 4 initialization methods: SearchF_lls (a; black), SearchF_nls (b; blue), HiLow (c; green) and HiLowDownhill (c; red), for varying noise levels (0% to 6%). The yellow vertical line indicates the ground truth solution, f = f* = 0.4. The mean estimated f across 100,000 experiments is marked with solid round marks, and the shaded regions mark one standard deviation. For the SearchF_lls and SearchF_nls methods, curves representing the minimal residual per f are plotted in gray (solid or dotted, for different noise levels). Note that ϵ_lls, the residual of the SearchF_lls method, is plotted in the logarithmic scale.

FIGURE 1.

Comparison of initialization methods for a single voxel experiment. As noise increases (13 noise levels, η = 0% to 6%), the SearchF_lls solutions (a; black) are biased towards f = 0, comparing with the ground truth (f* = 0.4; yellow line). The proposed SearchF_nls solutions (b; blue) are not biased. The proposed HiLowDownhill solutions (c; red) have lower residuals compared to HiLow (c; green). Note that the HiLow solutions are not accurate (high residuals) and biased to lower f, even for no noise. Round marks represent the mean estimated f across 100,000 experiments and regions shaded gray represent one standard deviation.

For the SearchF_lls method in fig. (1a), we observe that as the noise level grows, the found f deviates more from the ground truth of f* = 0.4 towards f = 0. Also, as noise increases, the curves become flatter with less clear minima, so that a large change in f results in a small change in the residual. When applying the improvement, we propose for the search algorithm, i.e., SearchF_nls, we see in Fig. (1b) that the curves maintain clear minima as the noise increases. More importantly, the estimated f are no longer biased towards f = 0.

Comparing the two additional initialization algorithms in Fig. (1c) we see that the HiLow-algorithm provides a consistent estimate of f as noise increases. The obtained f is, however, under-estimated. This bias also appears in the noiseless condition, where the non-zero residual suggests that the estimated D are biased as well. In comparison, also in the noiseless condition, the iterative HiLowDownhill-algorithm estimates f = f* well with ϵ_nls = 0. As noise increases, we see that the mean estimated f using the HiLowDownhill-algorithm remains close to the ground truth. Finally, for each noise level, the residuals of the HiLowDownhill-algorithm are smaller than those for the HiLow-algorithm.

4.2 |. The Synthetic Grid Dataset

The grid dataset allows investigating a range of different combinations of f values and D shapes and orientations for varying noise levels. Fig. (2a) plots the ϵ_nls residuals (calculated from each algorithm’s solution using Eq. (5)) to compare the performance of the different initialization methods, as well as the performance of subsequent non-linear minimization for each method. To directly compare the different methods with our proposed HiLowDownhill initialization, we subtract the HiLowDownhill residuals from the residuals of other methods in Fig. (2b). We observe that HiLowDownhill approach consistently outperforms all other methods considered, across all noise levels. In fact, the HiLowDownhill initialization outperforms the non-linear refined results of all other methods. We can also see that the performance of the final non-linear minimization highly depends on the selected initialization. The additional non-linear step had the most effect on the SearchF_lls initial values, reducing residuals by 20% for low noise and up to 32% for high noise. The non-linear step had the least effect on the HiLowDownhill initial values, improving residuals by 2% or less across all noise levels.

FIGURE 2.

Comparison of the ϵ_nls residuals of each algorithm for increasing noise levels of the Synthetic Grid dataset. The solid lines indicate the residuals of the initialization, and the dashed lines indicate the residuals following the non-linear minimization. The proposed HiLowDownhill initialization is very similar to the final non-linear result, and consistently outperforms other methods, as well as the non-linear steps following all other initializations.

4.3 |. The Simulated Brain dataset

As opposed to the grid dataset, the simulated brain considers a more realistic distribution of f and D values. To better understand in which ranges of f and D we may expect more bias for the different methods, we plot in Fig. (3) residuals and fitting differences from ground truth averaged over ranges of the ground truth f*. These plots are presented for low (SNR 33) and medium (SNR 20) noise levels (representative f and FA maps resulting from the different methods can be found in the Supplementary Material). We did not include SearchF_lls in the plots, since at these noise levels the method fails and estimates f ≈ 0 initial solutions in the vast majority of voxels (i.e., f < 0.05 in 88.9% and 99.9% of the voxels respectively).

FIGURE 3.

Comparison of initialization methods and their non-linear refinement over the Simulated Brain data. Voxels are grouped into 25 bins according to the ground truth value (f*). Plotted are (a) the mean ϵ_nls residuals, (b) fractional free-water difference f − f*, and (c) the tensor fractional anisotropy (FA) difference i.e. FA(D) − FA(D*), for two noise levels: low noise of SNR 33 (left column) and medium noise of SNR 20 (right column). Standard deviation for all plots can be found in Supplemental Material.

The HiLowDownhill and SearchF_nls methods provided similar initialization in terms of residuals, except that HiLowDownhill had better residuals in the large f* range (f* > 0.75 with low noise and f* > 0.65 with medium noise), where the SearchF_nls resulted with extreme under-estimation of f. In the low f* range (f* < 0.125 in the low noise and f* < 0.25 in the medium noise), SearchF_nls had slightly better residuals.

The non-linear refinement offered small improvements over the initializations for HiLowDownhill and SearchF_nls. For these two methods, the refinement appears to converge to very similar solutions for all but the highest f*.

The HiLow initialization had worse residuals than the other two approaches. The non-linear refinement improved the residuals especially for the lower f* range. As expected by the theoretical analysis, we see that HiLow generally underestimates f. Interestingly, in the very low range of f*, this causes the HiLow method to estimate the best f values, compared with the other methods which overestimate f at that range. Also of note, the FA estimations of the HiLow approach were generally the closest to the ground truth.

In terms of mean run-time, SearchF_lls run-time was 38.21s, the SearchF_nls run-time was 44.67s, HiLow run-time was 0.45s, and HiLowDownhill run-time was 8.76s.

4.4 |. In vivo Brain dataset

As a final comparison, we observe the initialization of the different methods on the acquired diffusion MRI data of the healthy volunteer. Since there is no ground truth, we visually compare the initialization results with a regularized non-linear fit⁶, that serves as a visual reference. Fig. (4) presents an example slice comparing the estimated f and FA of the initialization methods (prior to the non-linear refinement). For visualization purposes, the FA in the figures is weighted by 1 − f. This weighting suppresses random FA values for very high f voxels, where the tensor term becomes arbitrary.

FIGURE 4.

An example slice from the In vivo Brain dataset, comparing the free-water (f) and FA maps obtained from the different initialization methods. A regularized non-linear fit is presented as a reference image.

As predicted from the simulated brain, we see that SearchF_nls underestimates f in the large f ranges, such as in the ventricles. The f estimates of HiLow and HiLowDownhill are visually similar, although in 51.6% of the voxels, HiLow had smaller f than HiLowDownhill, compared to 28.4% voxels with larger f. The remaining 20% of voxels had identical f between the two methods, which occurs when the first step of HiLowDownhill does not improve the initialization provided by HiLow. Comparing the FA maps, we see that the three methods provide very similar results. The HiLowDownhill method tends to have slightly higher FA values than the HiLow method, and the FA image appears slightly smoother.

Fig. (5) provides a more quantitative comparison between the methods for this dataset and for the simulated brain dataset by plotting the mean residual (ϵ_nls) for each method before and after the non-linear refinement. We note that residuals in this experiment reflect the difference between the modeled signal and the acquired signal, which might be noisy, and therefore may not necessarily reflect the optimal solution. HiLowDownhill provides the lowest residuals and has similar residuals with or without the refinement step. In fact, the HiLowDownhill method provides slightly better residuals than the regularized non-linear fit image, which might be explained by the regularization that is applied on that image, which can force the results to deviate from the underlying signal in favor of the additional constraints. The differences between the methods were similar for the in-vivo brain and the simulated brain, suggesting that the in vivo brain had noise levels in between the low and medium noise levels used for the simulated brain.

FIGURE 5.

The mean voxelwise ϵ_nls residuals of the algorithms averaged across the brain. The legend indicates the initialization result colors, and the lighter hues (to the right of each column) are used for the non-linear refinement results.

5 |. DISCUSSION

In this work we provide a comprehensive assessment of different initialization approaches that help the fitting of a two-compartment free-water model. We show analytically that the previously proposed free-water LLS estimation is biased towards f = 0. We show that by modifying the estimation to use a non-linear criterion (NLS), the bias is removed. Finally we propose the HiLowDownhill initialization approach that improves the HiLow approach and circumvents the brute force approach used by the search algorithm applied to the LLS and NLS criteria by leveraging the different information embedded in high and low b-shells. We note that the search based approaches are limited to a pre-defined grid, and despite their brute force approach, are not guaranteed to find an optimal solution that is likely in-between the defined grid. In fact, we show that the proposed HiLowDownhill approach provides superior or equal quality of fit in most ranges, compared to all other methods. Non-linear refinement remains an important supplement for the initialization step, although we demonstrate that the HiLowDownhill approach provides initial results that are very close to those obtained by the non-linear refinement, suggesting that the initialization on its own may be considered as a good approximation of the final result.

Our analytic and simulated results clearly demonstrate that the SearchF_lls algorithm performs poorly in a noisy experiment, resulting with a bias towards f = 0. This is of importance since the LLS algorithm was used in previous publications^18,29,30, suggesting that the results provided in these publications were initialized by underestimated f. By simply replacing the cost function this bias can be avoided. In our proposed approaches, the estimation of a tensor D from a given f is still performed using LLS. However, this step could potentially be replaced by other estimators that may not be as fast, but that are more accurate or less biased, than the LLS estimator. Furthermore, the proposed initialization methods explicitly minimize the residuals (i.e., difference of the model’s predicted signal from the acquired signal). However, In noisy data, minimizing the residuals may cause overfitting, which means that fitted parameters that reduce residuals are not necessarily closer to the ground truth (as can be seen, for example, in Fig. 3). Nevertheless, minimizing residuals remains a preferred way to estimate a given model’s performance.

By utilizing the additional information in the multi-shell data, the HiLow and HiLowDownhill approaches allow a direct and fast estimation of the model parameters. We can see from our analyses that the HiLow approach is not optimal in terms of the residual fit. However, once the iterative HiLowDownhill approach is applied, the fit is stabilized and dramatically improved. This should not be surprising, since HiLowDownhill, by design, always provides equal or better residuals than the HiLow approach, with the exception of voxels where the HiLow approach resulted with a negative tensor, in which case, the HiLowDownhill approach will map this negative tensor to a positive tensor that might have higher residual than the initial negative tensor. Between the two proposed methods, HiLowDownhill provides a better or equivalent fit compared with SearchF_nls, and since HiLowDownhill is faster to compute, it can be recommended as a preferred initialization approach.

With respect to positive definite tensors, it is important to note that the StayPositive approach we provided here to maintain positive tensors is a heuristic, and by no means should it be regarded as the optimal approach for dealing with negative tensors. Nevertheless, StayPositive provides a simple way to convert any negative tensor to a positive one, while avoiding some of the issues associated with more complicated approaches^25,31. The problem of negative tensors is shared by the two-compartment free-water model and by the DTI model. However, with a naïve fitting approach, negative tensors are more likely to appear in the free-water model fit (or any other multi-compartmental model), because the elimination of the isotropic compartment effectively reduces the eigenvalues of the tensor compartment, bringing them closer towards zero. One limitation of the suggested StayPositive-pseudocode implementation is that it will continue searching until the minimal eigenvalue is just slightly larger than zero, essentially returning a (near) singular tensor. Further approaches for dealing with negative eigenvalues are needed.

While our derivations support any weighting term for the LLS fit, in our experiments we used equal weights (i.e., w_i = 1), which may not be optimal for non-biased estimation of the tensor model. Optimally, weighted least squares is useful to reduce bias that may be introduced by non-symmetric noise distribution, such as Rician noise, and especially so in low SNR³². Selecting ${\text{w}}_{\text{i}}={\text{s}}_{\text{i}}^2$ is expected to improve the fit by suppressing low signal, where noise is amplified by the logarithmic transform^3,19. However, optimal selection of weights is still under debate^19,33, with limitations and pitfalls that involve inappropriate selection of weights³⁴. The implications of weighted least squares on multi-b and multi-compartmental models is less trivial, since the noise distribution may depend on the b-value³⁴, and on the intrinsic diffusion profile of each fitted compartment. More elaborate weightings that take into account the relative b-values across shells, as well as potentially the number of samples in each shell, may be more appropriate for the fitting approaches suggested here. Further studies are needed to explore the effect of different weighting selection on least squares fit of multi-b and multi-compartmental fit, and on the selection of optimized gradient schemes. In addition, other emerging non-linear least square estimation approaches (e.g., ³⁵) may be considered to remove bias introduced by non-symmetric noise distributions.

Multi-shell data is becoming increasingly common, with off-the-shelf support of some of the leading MRI vendors. Accordingly, most recent consortia and large clinical studies opt to choose multi-shell data over single-shell data. While the Free-Water Imaging model can be estimated from a single-shell acquisition⁶, the acquisition of multi-shell data is highly recommended. This is since fitting single-shell data is extremely dependent on regularization constraints, which in turn may over-smooth or reduce subtle image variabilities¹⁷. Nevertheless, regularization may be beneficial for a more accurate fit also for multi-shell data. Therefore, future work is encouraged to combine the initialization methods proposed here with more advanced fitting approaches that include statistical priors or regularizers (e.g.,^13,10).

Despite it being 25 years since the inception of DTI, the tensor model remains a robust representation of microstructural features that is especially useful for the identification of brain abnormalities³⁶. The two-compartment model enhances the DTI model by eliminating effects of free-water. However, the tensor compartment is still modeled by a diffusion tensor. With the availability of multi-shell data, more complicated models that aim to identify specific microstructural changes become available. These methods, including diffusion basis spectrum imaging ⁸, neurite orientation dispersion and density imaging⁹, multiple fascicle models¹¹, and diffusion kurtosis imaging³⁷, also benefit from the elimination of the free-water compartment that is not expected to contribute to the microstructural profile of brain tissue. Robust identification of the free-water compartment could thus contribute as a preprocessing step for the fitting procedures of these abovementioned methods.

The two compartment model may further be improved by including additional fast diffusing compartments to account for pseudo-diffusion in micro-capillaries³⁸, and by accounting for relaxation times that are distinctly different between free-water and tissue^39,40. Such approaches require additional low b-value shells, or multiple echo times, which makes the acquisition considerably longer, and less feasible for clinical studies. Nevertheless, advancement in multi-band acquisitions⁴¹, as well as novel pulse sequences⁴² show promise for obtaining signal that may feasibly obtain more accurate estimation of the free-water compartment.

In conclusion, the fast initialization approaches proposed here improve previous initialization approaches, and provide initial estimations that in many cases are close to the optimal solutions. Therefore using the initialization approaches improves the quality of the Free-Water Imaging model fit, and provides a fast alternative for estimating the model parameters.

⁰Abbreviations

MRI

Magnetic Resonance Imaging

DTI

Diffusion Tensor Imaging

FWI

Free-Water Imaging

LLS

Linear Least-Squares

NLS

Non-linear Least-Squares

SNR

signal-to-noise ratio

APPENDIX

A THE SYNTHETIC GRID

To compare the initialization approaches over a range of D and f combinations we created a synthetic phantom comprising a volume of 50 × 50 × 50 voxels.

A single 50 × 50 slice contained a collection of diffusion tensors with varying orientation and anisotropy, and the arrangement of the tensors was identical for each slice while the f value changed between slices. The tensors varied by rotation, where tensors rotated in each voxel by 7.2 degrees along the x-axis to complete exactly 2 × 180 degrees rotations. Along the y-axis, the anisotropy varied linearly and in a symmetric fashion along the center line. This variation created an “S” shape. At the center line the eigenvalues are the most anisotropic with ${\lambda }_{\text{i}}=\{1.6,0.4,0.4\}\cdot {10}^{-3}{\text{mm}}^2/\text{s}$ and at the edges, y ∈ {1, 50}, the eigenvalues are isotropic with λ_i = 0.8 · 10⁻³mm²/s. Following¹⁸ all tensors had a trace of 2.4 · 10⁻³mm²/s. Along the z-axis, the free-water component f* varied linearly with z between 0 and 1, where f increased by $\frac{1}{50}$ in each slice. This synthetic grid yielded a uniform distribution of tensors and f values.