Performance of ODF-Fingerprinting with a biophysical multicompartment diffusion model

Abstract

Purpose:

Orientation Distribution Function (ODF) peak finding methods typically fail to reconstruct fibers crossing at shallow angles below 40 degrees, leading to errors in tractography. ODF-Fingerprinting (ODF-FP) with the biophysical multicompartment diffusion model allows for breaking this barrier.

Methods:

A randomized mechanism to generate a multidimensional ODF-dictionary that covers biologically plausible ranges of intra- and extra-axonal diffusivities and fraction volumes is introduced. This enables ODF-FP to address the high variability of brain tissue. The performance of the proposed approach is evaluated on both numerical simulations and a reconstruction of major fascicles from high- and low-resolution in vivo diffusion images.

Results:

ODF-FP with the suggested modifications correctly identifies fibers crossing at angles as shallow as 10 degrees in the simulated data. In vivo, our approach reaches 56% of true positives in determining fiber directions, resulting in visibly more accurate reconstruction of pyramidal tracts, arcuate fasciculus, and optic radiations than the state-of-the-art techniques. Moreover, the estimated diffusivity values and fraction volumes in corpus callosum conform with the values reported in the literature.

Conclusion:

The modified ODF-FP outperforms commonly used fiber reconstruction methods at shallow angles, which improves deterministic tractography outcomes of major fascicles. In addition, the proposed approach allows for linearization of the microstructure parameters fitting problem.

1 ∣. INTRODUCTION

Visualizations of White Matter (WM) fibers are reconstructed in vivo from Diffusion Weighted Images (DWIs) through tractography. Numerous studies have shown agreement between tractography and neuroanatomy when identifying major fascicles [1, 2, 3, 4, 5], although — at the voxel level — reconstruction of WM fibers often remains uncertain [6, 7].

Tractography algorithms typically identify spatial orientation of fibers by seeking maxima of Orientation Distribution Functions (ODFs) [8, 9, 10, 11]. The latter are probability distributions of preferred diffusion directions approximated from a finite set of samples using one of the following approaches: The so-called diffusion ODFs (dODFs) [12] are calculated as the discrete Fourier transform of a signal acquired with a dense multi-shell sampling scheme. From single shell acquisitions, q-ball ODFs (qODFs) [13] can be estimated with a spherical Radon transform. Fiber ODFs (fODFs) are obtained by deconvolving the contribution attributed to a single fiber from the diffusion signal [8, 14]. Alternatively, fiber directions can be reconstructed by sampling a probabilistic model of acquired data [15, 16] or training convolutional neural networks [17, 18]. Nonetheless, most of current approaches fail to reconstruct fibers crossing at shallow angles below 40° due to limited angular resolution and ODF peak width [6, 7]. Even though the angular resolution can be augmented by increasing the number of gradient directions, it does not effectively improve the ability to resolve crossing fibers at shallow angles because of blurring artifacts [14]. ODF-Fingerprinting (ODF-FP) [19] alleviates these issues by replacing the commonly used fiber identification methods with pattern matching. Here, ODFs calculated from DWIs — either as dODFs, qODFs, or fODFs — are compared with elements of a synthetically generated ODF-dictionary, aiming to find the closest match and thus look up the underlying fiber structure that likely contributed to the given signal. Through this, we use all information embedded in ODFs to empower fiber reconstruction rather than rely solely on peak finding. Our earlier study [19] has shown that ODF-FP equipped with an ODF-dictionary generated with the multi-tensor model [20] successfully reconstructed multiple fibers crossing at shallow angles.

In this work, we extend the applicability of ODF-FP by harnessing a biophysical multicompartment diffusion model [21] to support fiber reconstruction efficacy. We construct multidimensional ODF-dictionaries that cover biologically plausible ranges of intra- and extra-axonal diffusivities and fraction volumes. This enables ODF-FP to address the high variability of brain tissue, thus allowing for more accurate tractography. We evaluate this approach both in silico and in vivo. In particular, we assess the reconstruction of cortical terminations of selected major fascicles using high- and low-resolution DWIs.

Beside improved identification of crossing fibers, our proposed modification of ODF-FP enables reconstruction of tissue microstructure through minimization of the cosine similarity between the measured and the generated ODFs rather than through time-consuming exploration of a high-dimensional parameter space [22, 23]. A similar concept was introduced by Daducci et al. [24] who showed the potential of linearization in accelerating microstructure reconstruction. Here, we compare our results with the state-of-the-art fitting algorithms [23, 25, 26]. Unlike the latter, ODF-FP manages to fit the multicompartment model with up to 3 intra- and extra-axonal components per voxel. The diffusivity values and fraction volumes found with our method in vivo conform with the values reported in literature [27, 28, 29].

A limiting factor in ODF-FP is the dictionary size needed for full sampling of the parameter space. In this work, we introduce a novel randomized approach, which allows for controlling the ODF-dictionary size manually. We test our randomized ODF-dictionaries using various simplifications of the multicompartment diffusion model to assess robustness of ODF-FP at different levels of complexity. Having that, we suggest an effective generation mechanism and input parameter ranges to provide guidance for using ODF-FP in practice.

Our results show that ODF-FP equipped with the biophysical multicompartment diffusion model is capable of producing detailed reconstruction of major fascicles, which is crucial for tractography-based surgical planning [30] and measurement of brain connectivity. Moreover, our modified ODF-FP has a potential to estimate the microstructure parameters in linear time leading to significant reduction in complexity of the fitting problem.

2 ∣. THEORY

In this section, we briefly formalize the concept of ODF-FP [19].

2.1 ∣. ODF-FP matching

Let $\mathbf{u}\in {\mathbb{R}}^3$ such that ∥u∥ = 1. For the scope of this paper, let us assume that

$$\text{ODF}(\mathbf{u})={\int }_0^{\infty }P(r\mathbf{u})r^{\alpha }dr,\alpha \in \{1,2\}$$ (1)

holds the probability P that an ensemble of water molecules diffuses in the direction u, integrating all displacement distances r > 0 [31]. The exponent α is controlled by a reconstruction method and will be specified later.

Consider a uniform k-point tessellation (k ≥ 3) of a unit hemisphere, which discretizes the set of possible directions $\{\mathbf{u}\in {\mathbb{R}}^3:\Vert \mathbf{u}\Vert =1\}$ onto k unit vectors {u₁, …, u_k}. Having these, we define a normalized k-point ODF-fingerprint (or simply ODF-fingerprint) as follows

$$\mathbf{x}=\frac{(\text{ODF}({\mathbf{u}}_1),\dots ,\text{ODF}({\mathbf{u}}_k))}{\sqrt{{\sum }_{i=1}^k\text{ODF}({\mathbf{u}}_i{)}^2}}\in {\mathbb{R}}^k,\Vert \mathbf{x}\Vert =1.$$ (2)

Let the matrix $\mathcal{D}\in {\mathbb{R}}^k\times {\mathbb{R}}^n$ hold a collection of n > 0 synthetically generated ODF-fingerprints sampled at u₁, …, u_k. We refer to $\mathcal{D}$ as ODF-dictionary. It is important to note that the columns of $\mathcal{D}=[{\mathbf{d}}_i{]}_{1\le i\le n}$ are ODF-fingerprints generated using an arbitrary diffusion model with known parameters.

Assume that x in Equation 2 is obtained from an acquisition of DWIs. A vector $\tilde{\mathbf{x}}\in {\mathbf{R}}^k$ defined as

$$\tilde{\mathbf{x}}=\text{arg}\underset{{\mathbf{d}}_i\in \mathcal{D}}{\max }{\mathbf{x}}^T{\mathbf{d}}_i$$ (3)

maximizes the cosine similarity between the ODF-fingerprint x of the acquired signal and the synthetically generated ODF-fingerprints d_i stored in the ODF-dictionary. In other words, $\tilde{\mathbf{x}}$ is the closest approximation of x among the elements of $\mathcal{D}$.

By approximating x with $\tilde{\mathbf{x}}$ (Figure 1), we gain three types of benefits due to the fact that $\tilde{\mathbf{x}}$ is generated with known parameters. First, we simply look up the number and orientations of fibers contributing to $\tilde{\mathbf{x}}$ instead of seeking the peaks of x explicitly. Second, we look up the diffusivities and other microstructure parameters of our assumed diffusion model without the need to explore the high-dimensional parameter space to find the best fit. Third, we can compensate for the blurring effect in ODFs [6, 7, 14] by using all information stored in x to reconstruct crossing fibers rather than rely solely on peak finding.

FIGURE 1.

Schematic illustration of a simplified hypothetical two-dimensional variant of ODF-FP applied for signal ODFs sampled at 36 gradient angles (pictured as gray radial lines). The surface plots show ODFs representing pairs of fibers crossing at arbitrary angles (10°, …, 80°), generated synthetically with SNR=10 (signal-to-noise ratio). The curves represent discretized input ODFs (blue) and their corresponding dictionary ODFs (red) found using pattern matching. This approach allows ODF-FP to identify fibers crossing at shallow angles below 40°, even when ODF peaks merge into one blob resembling a single fiber configuration.

The robustness of our approach relies on accuracy of $\tilde{\mathbf{x}}$ as the approximation of the “true” ODF represented by x. One notable disrupting factor here is the presence of noise in x which may lead to overfitting of the ODF-dictionary items and a generation of spurious ODF peaks [19]. We account for that by incorporating into Equation 3 an adaptation of Akaike Information Criterion [32] defined as follows

$$\tilde{\mathbf{x}}=\text{arg}\underset{{\mathbf{d}}_i\in \mathcal{D}}{\max }\left(\log {\mathbf{x}}^T{\mathbf{d}}_i-2N_i\cdot \lambda {\sigma }^2\right),$$ (4)

where N_i > 0 is the number of crossing fibers contributing to d_i, σ² is the voxel-wise variance of noise, and λ > 0 is the regularization parameter.

2.2 ∣. ODF-dictionary

Consider an arbitrary voxel of a diffusion-weighted image that contains N > 0 crossing fibers. Without loss of generality, we choose a Cartesian coordinate system for this voxel ensuring that the dominating fiber is oriented along the z-axis, i.e. the main diffusion direction is (0, 0, 1). Note that for any unit vector v = (v₁, v₂, v₃) ≠ (0, 0, 1) this assumption can be met by applying the rotation defined as

$$R_{v\to (0,0,1)}=I+A+A^2\frac{1-v_3}{v_1^2+v_2^2},\text{where}A=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill -v_1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -v_2\hfill \\ \hfill v_1\hfill & \hfill v_2\hfill & \hfill 0\hfill \end{array}\right].$$ (5)

Assume that the uniform k-point tessellation, introduced in Section 2.1, includes the orientation vector (0, 0, 1). Thus, our ODF-dictionary needs only this one direction to represent a single-fiber voxel. For N = 2, we can distinguish k − 1 configurations of non-collinear crossing fibers in our discrete k-point sampling system. For N = 3, there are (k − 1) (k − 2) possibilities, etc. As a consequence, an ODF-dictionary with N ≤ 3 should contain fractions of items representing 1, 2, and 3 fibers in the proportion 1 : k − 1 : (k − 1) (k − 2) to account for the distribution of directions.

Depending on the diffusion model assumed a priori, the ODF-dictionary must cover its respective space of parameters. This can be done exhaustively through enumeration of discretized parameter configurations [19]. However, the size of the ODF-dictionary grows exponentially with the number of Degrees Of Freedom (DOF) in such an approach, which aggravates the memory complexity (Figure 2). Note that data collections above 10¹² bytes = 1TB may pose considerable storing problems, which makes them highly impractical as ODF-dictionaries. To alleviate this, we propose to draw the configurations of parameters randomly, allowing for a full flexibility of the dictionary size n.

FIGURE 2.

Enumerated ODF-dictionary sizes grow exponentially with the number of samples taken per parameter of the diffusion model (illustrated on x-axes). The plots show memory requirements (in bytes) for storing enumerated ODF-dictionaries of the full model and its three simplifications (defined in Subsection 3.4) for N ≤ 2 fibers per voxel (left plot) or N ≤ 3 (right plot), assuming 321-point tessellation. The respective memory sizes of the randomized ODF-dictionaries with n = 10³, 10⁴, 10⁵, 10⁶ elements are shown for reference as green horizontal lines.

3 ∣. METHODS

In this work, we evaluate the performance of ODF-FP equipped with the biophysical multicompartment diffusion model, emphasizing the identification of crossing fibers and the determination of crossing angles. For comparison with other approaches, we computed the respective ODF peaks using the local maximum search (Local max) from DSI Studio [33], the Newton search based on the MRtrix3 [34] tool sh2peaks, the Constrained Spherical Deconvolution (CSD) [8, 14] with the response function estimated using dwi2response (MRtrix3) with default settings, and the Probabilistic approach of FSL Bedpostx [15, 16].

3.1 ∣. Biophysical multicompartment diffusion model

We used numerical simulations for too distinct purposes — to generate in silico experimental data and to produce ODF-dictionaries. In either case, we applied a biophysical multicompartment model of WM microstructure [21] defined as

$$S_g(b)=S(0)\cdot \left[p_{iso}{e}^{-b{D}_{iso}}+\sum _{i=1}^N{p}^{(i)}{\kappa }^{(i)}(b,\mathbf{g}\cdot {\mathbf{n}}^{(i)})\right],$$ (6)

where S(0) is the signal without diffusion encoding (b = 0). The contribution of every i-th fiber (i = 1, …, N) is

$${\kappa }^{(i)}(b,\mathbf{g}\cdot {\mathbf{n}}^{(i)})=f^{(i)}{e}^{-b{D}_{a,\parallel }^{(i)}(\mathbf{g}\cdot {\mathbf{n}}^{(i)}{)}^2}+\left(1-f^{(i)}\right)e^{-b{D}_{e,\parallel }^{(i)}(\mathbf{g}\cdot {\mathbf{n}}^{(i)}{)}^2-b{D}_{e,\perp }^{(i)}\left(1-(\mathbf{g}\cdot {\mathbf{n}}^{(i)}{)}^2\right)},$$ (7)

where n⁽ⁱ⁾ the fiber orientation, g is the direction of the diffusion-encoding gradient, $D_{a,\parallel }^{(i)}$ is the intra-axonal diffusivity and $D_{e,\parallel }^{(i)}$, $D_{e,\perp }^{(i)}$ are the extra-axonal diffusivities. In the remainder of this paper, we will refer to this model as the full model.

To ensure sufficient coverage of the model parameters, we drew the following values randomly using a uniform distribution over supersets of anatomically plausible ranges [27, 28, 29]:

• volumes of free water p_iso ≥ 0 and neurites p⁽ⁱ⁾ ≥ 0.1, ensuring that $p_{iso}+{\sum }_{i=1}^N{p}^{(i)}=1$,

• intra-axonal fraction sizes f⁽ⁱ⁾ ∈ [0, 0.8],

• diffusivities of free water D_iso ∈ [2, 3] · 10⁻⁹ m²/s, intra-axonal $D_{a,\parallel }^{(i)}\in [1.5,2.5]\cdot {10}^{-9}{\mathrm{m}}^2∕\mathrm{s}$, and extra-axonal $D_{e,\parallel }^{(i)}\in [1.5,2.5]\cdot {10}^{-9}{\mathrm{m}}^2∕\mathrm{s}$, $D_{e,\perp }^{(i)}\in [0.5,1.5]\cdot {10}^{-9}{\mathrm{m}}^2∕\mathrm{s}$.

We allowed for some variability in the number of fibers per voxel, yet we restricted it to N ≤ 3, as suggested by Jeurissen et al. [36].

Finally, let us point out that a “uniform” distribution of fiber orientations n⁽ⁱ⁾ ∈ [−π, π] × [−π/2, π/2] can be defined in numerous ways. Here, we considered a tessellation of a unit hemisphere with k = 321 vertices (ensuring angular resolution of 2.3°) and we drew the directions n⁽ⁱ⁾ randomly among these vertices. Note that our tessellation purposefully surpassed the angular resolution of considered DWIs aiming to improve the rendition of ODFs and to help identify their shapes.

3.2 ∣. Synthetic data experiments

For the experiments in silico, we chose a clinically plausible set of b-values, b = 1000, 2000, 3000 s/mm², sampled at 90 radial lines (i.e. having the same gradient directions at every b-shell). Next, we computed the corresponding ODFs using Radial Diffusion Spectrum Imaging (RDSI) [37] reconstruction with the exponent α = 2 (Equation 1) to reduce spurious peaks in ODFs. In each experiment, our data sets comprised 100000 voxels with added Rician noise at the signal-to-noise ratios: SNR = 50, 20, 10.

3.3 ∣. In vivo diffusion data

We used minimally preprocessed DWIs [38] of a healthy subject from the Human Connectome Project (HCP) data set provided by the Washington University in Saint Louis and the University of Minnesota [39]. The images were acquired on a Siemens 3T Skyra scanner using a 2D spin-echo single-shot multiband EPI sequence with a multi-band factor 3 and monopolar gradient pulse. The spatial resolution was 1.25 mm isotropic, TR = 5500 ms, TE = 89.50 ms. The b-values were 1000, 2000, 3000 s/mm² with 90 acquisition directions per shell, and 6 images at b = 0.

Due to the lack of ground truth in vivo diffusion data, we downsampled our DWIs to the 2.5 mm isotropic resolution with mrresize command from MRtrix3 and used the original high-resolution data as reference. For calculating ODFs, we used Generalized Q-space Imaging (GQI) [40] reconstruction with the exponent α = 1 (Equation 1).

3.4 ∣. ODF-dictionary generation

We generated our ODF-dictionaries randomly with the uniform distribution over the parameter ranges defined in Subsection 3.1. The sampling of the q-space and b-values as well as the ODF reconstruction method (either RDSI or GQI) mirrored the ones of the input diffusion signal. For the remaining parameters, we considered combinations of the following:

• ODF-dictionary size n = 10³, 10⁴, 10⁵, or 10⁶,

• number of crossing fibers per voxel N ≤ 2 or N ≤ 3.

The ODF-dictionaries contained fractions composed of 1 or 2 fibers (for N ≤ 2) and 1, 2, or 3 fibers (for N ≤ 3) in the proportions reflecting the combinations of angular configurations, as introduced in Subsection 2.2. Particularly, our tessellation of a unit hemisphere determined 321 distinct orientations, which defined the proportion 1 : 320 : 102080. This means that for selecting the number of fibers in a given dictionary item, a random integer was chosen among 1 and 102401 (since 1 + 320 + 102080 = 102401). Hence, the probability of choosing 1 fiber was 1/102401,2 fibers was 320/102401, and 3 fibers was 102080/102401.

Table 1 shows the theoretical breakdown of randomly generated ODF-dictionaries for the considered dictionary sizes n. Note that the configuration with N ≤ 2 fibers per voxel required as little as 10³ dictionary items to cover the parameter space with at least 3 samples with 1 fiber per voxel and 997 with 2 fibers. For N ≤ 3, the ODF-dictionary had to contain more than 10⁵ elements in order not to compromise the angular accuracy. Otherwise, there would be only 31 samples with 2 fibers (for n = 10⁴) or even 3 samples (for n = 10³) as indicated in Table 1, which means that most of the directions would be omitted.

TABLE 1.

Theoretical breakdown of randomly generated ODF-dictionaries of the sizes n = 10³, 10⁴, 10⁵, and 10⁶ (in rows) for 321-point discretization of the set of fiber directions. The groups of columns represent the variants with N ≤ 2 and N ≤ 3 fibers per voxel.

	breakdown of ODF-dictionary items
dictionary size	# fibers per voxel ≤ 2		# fibers per voxel ≤ 3
	N = 1	N = 2	N = 1	N = 2	N = 3
103	3	997	1	3	996
104	31	9969	1	31	9968
105	311	99689	1	312	99687
106	3,115	996885	10	3125	996865

For the ODF-FP matching formula (Equation 4), we approximated the voxel-wise noise variance σ² of the input signal using the technique described by Aja-Fernández et al. [41]. With that in hand, we chose an empirical regularization parameter $\lambda =5∕(1000\cdot \text{median}({\sigma }_{img}^2))$, where $\text{median}({\sigma }_{img}^2)$ is the combined image-wise median of noise variance from all voxels.

Diffusion model simplifications

Our multicompartment diffusion model, introduced in Subsection 3.1, has 2 + 7N DOF for any given N, i.e. 23 at the maximum (N = 3). Due to the previously reported uncertainty when fitting such high-dimensional models of diffusion [42], we added to our study the following three simplifications of the full model:

1° equal contributions of all fibers, or equal fibers for short (6 + 3N DOF):

$$D_{a,\parallel }^{(i)}=D_{a,\parallel },D_{e,\parallel }^{(i)}=D_{e,\parallel },D_{e,\perp }^{(i)}=D_{e,\perp },f^{(i)}=f$$ (8)

2° equal intra-axonal contributions only, or intra only (3 + 3N DOF):

$$D_{a,\parallel }^{(i)}=D_{a,\parallel },f^{(i)}=1$$ (9)

3° equal extra-axonal contributions only, or extra only (4 + 3N DOF), which boils down to the multi-tensor model:

$$D_{e,\parallel }^{(i)}=D_{e,\parallel },D_{e,\perp }^{(i)}=D_{e,\perp },f^{(i)}=0$$ (10)

Degenerated ODF-dictionaries

Aiming to evaluate the impact of mismatched ODF-dictionaries on the overall fiber identification accuracy of ODF-FP, we created a set of degenerated ODF-dictionaries where a certain diffusivity was chosen as constant — either D_a,∥ = 2, D_e,∥ = 2, or D_a,⊥ = 1 [ · 10⁻⁹ m²/s ]. Note that similar assumptions were made in Neurite Orientation Dispersion and Density Imaging (NODDI) [43] or Free Water Elimination (FWE) [44], where diffusivities were fixed at arbitrary values. Here, we used central values of plausible diffusivity ranges defined in Subsection 3.1.

A comparison between fibers reconstructed with the regular and the degenerated ODF-dictionaries helped us observe the potential robustness of ODF-FP when dealing with input data from outside the scope of an ODF-dictionary. Note that such situations may occur in practical scenarios of distorted DWIs or pathological tissue.

Biophysical plausibility

In order to ensure biophysical plausibility of our ODF-dictionaries generated for in vivo experiments, we asserted the inequality relation between intra- and extra-axonal diffusivities D_a,∥ ≥ D_e,∥ advocated by Jelescu et al. [45] and supported by earlier studies [46, 47, 48, 28, 49, 50, 51].

3.5 ∣. Evaluation

Crossing angle accuracy

The assessment of the crossing angle accuracy required a dedicated preprocessing of the simulated data to ensure a fair comparison. For this, we generated 100000 voxels with N = 2 crossing fibers and grouped them by fiber crossing angles α_true ∈ {1°, …, 90°} into 9 intervals: 1–10°, 11–20°,…, 81–90°. We required that each such interval be equally represented, i.e. it contained approximately 100000/9 data samples. Having these, we computed the rates of correct angles per interval, using the acceptance criterion ∣α_true − α_found∣ ≤ ε with error tolerance levels ε ∈ {10°, 15°, 20°}.

Fiber identification

In this experiment, we focused on identification of numbers of fibers per voxel. To this end, we generated a synthetic data set of 100000 voxels with N ≤ 3 and grouped them into fractions with N = 1, 2, or 3 crossing fibers. We ensured that each such fraction contained approximately 100000/3 data samples. Any fiber found by our tested algorithms was considered correct if the angle between this fiber and the ground truth did not exceed 30°. Note that here and in another experiment with N ≤ 3 fibers per voxel, we increased the error tolerance to account for the blurring effect observed in ODFs representing more than 2 crossing fibers.

Tractography assessment

In the case of in vivo diffusion data, we applied a qualitative and quantitative evaluation. The former comprised of a visual inspection of pyramidal tract (PyT), arcuate fasciculus (AF), and optic radiations (OR) in both high- and low-resolution data. We reconstructed the fascicles using the DSI Studio deterministic tractography algorithm (the winner of the ISMRM 2015 Tractography Challenge [5]) and the built-in automatic atlas-based virtual dissection [33]. The stopping criteria were defined by the adaptively chosen Quantitative Anisotropy (QA) threshold [52], the differential tracking threshold 0.2, and the angular threshold 60°. For each fascicle, we produced 10000 streamlines using the 4th-order Runge-Kutta integration [9], with streamline lengths ranging between 30 and 300 mm. The step size was equal to half of the image resolution, i.e. 0.62 mm in the high- and 1.25 mm in the low-resolution variants.

Our qualitative approach — due to the lack of ground truth in vivo data — targeted robustness in identifying fiber directions under the low-resolution regime, as we proposed earlier [19]. To this end, we ran our in-house Matlab^® code to compare the fibers found in each voxel of the low-resolution image with the corresponding 2×2×2 = 8 voxels of the high-resolution one. Whenever a low-resolution ODF peak matched with a high-resolution peak (i.e. the angle between them did not exceed 30°), we considered it a true positive (TP). Otherwise, we counted it as a false positive (FP). All the high-resolution peaks without homologues in the corresponding low-resolution voxel were classified as false negatives (FNs). For such obtained absolute numbers of TPs, FPs, and FNs, we computed the rates of TPs and FPs defined as

$$\mathsf{True}\mathsf{positive}\mathsf{rate}=\frac{\#\mathsf{TP}}{\#\mathsf{TP}+\#\mathsf{FP}+\#\mathsf{FN}},$$ (11)

$$\mathsf{False}\mathsf{positive}\mathsf{rate}=\frac{\#\mathsf{FP}}{\#\mathsf{TP}+\#\mathsf{FP}+\#\mathsf{FN}}.$$ (12)

Note that the above assessment exploited the coarse-graining effect that we induced by downsampling the DWIs. In particular, certain fiber directions that were identifiable at the high-resolution level became more difficult to reconstruct (due to aggregation of the signal) at the coarse-grain level. Such an approach is frequently used in assessment of robustness of MRI signal reconstruction, e.g. in super-resolution techniques [53].

Microstructure parameters fitting

The last set of experiments was focused on determination of diffusivities D_a,∥, D_e,∥, D_e,⊥ and volumes p_iso, p⁽ⁱ⁾, $f_{in}^{(i)}$. We compared our ODF-FP approach with two frequently used algorithms implemented in dmipy [22] — namely MIX optimizer [23] and Brute2Fine [25, 26] — on the synthetic data set. Due to relatively large memory complexity of the latter two approaches, we were able to run them with N ≤ 2 fibers per voxel variant only. For the in vivo diffusion data, we used a more realistic assumption that N ≤ 3 [54, 55, 56]. Here, we compared our findings in the Regions of Interest (ROIs) defined in corpus callosum (genu and splenium) and all WM voxels with the results published in the literature.

4 ∣. RESULTS

Synthetic data results

ODF-FP reached highest accuracy among all tested methods when identifying challenging crossing angles below 40° in the synthetic diffusion data. Figure 3 presents the results for the ODF-dictionary containing 10⁵ elements generated with the full model at N ≤ 2. The accuracy of the probabilistic approach grew steadily with increasing crossing angle and reached the level of ODF-FP above 40°. On the contrary, CSD, Local max, and Newton search were visibly less accurate than the other two methods regardless of the crossing angle. Also note that the presence of Rician noise caused a small degradation of results for all tested methods (Figure 3).

FIGURE 3.

ODF-FP reached highest accuracy among all tested methods when identifying crossing angles below 40° in the synthetic data experiment. The plots show percent rates of correctly identified crossing angles under the error tolerance ε = 10°, 15°, 20°. The ground truth data are grouped into 9 intervals: 1 – 10°, 11 – 20°,…, 81 – 90°. The three variants of additive Rician noise with SNR=50, 20, 10 are presented in rows. In all cases, the ODF-dictionary contained 10⁵ elements generated with the full model. The number of fibers per voxel was N ≤ 2.

The role of the ODF-dictionary size turned out to be negligible in the experiment with two crossing fibers per voxel. For all considered variants (i.e. 10³, 10⁴, 10⁵, and 10⁶), we obtained comparable angular precision below 10° at SNR = 50, decreasing proportionally to 25° at SNR = 10 (Figure 4).

FIGURE 4.

The ODF-dictionary sizes (left column) and most of the ODF-dictionary simplifications (right column) had little or no impact on the angular precision of ODF-FP in the synthetic data experiment. The box plots show the crossing angle reconstruction errors. The three variants of additive Rician noise with SNR=50, 20, 10 are presented in rows. All ODF-dictionaries were generated with the number of fibers per voxel N ≤ 2.

The parameter intervals used in the ODF-dictionary generation had a minor impact on the crossing angle reconstruction accuracy (Figure 4). Most of our model simplifications — even the cases with constant diffusivities — resulted in reconstruction errors below 10° at SNR = 50 and maintained similar robustness to noise. One notable exception was the extra only simplification which was the sole variant that led to significant accuracy loss (Figure 4).

ODF-FP and the FSL-based probabilistic approach were visibly more robust in the fiber number determination task than the other tested algorithms (Figure 5). ODF-FP correctly identified almost all the voxels with a single fiber and more than a half of the 2- and 3-fiber configurations regardless of the noise level. The FSL tool performed better on 3 fibers, although was much less accurate in the single fiber case. Local max and CSD failed to recognize most of the 3-fiber configurations, identifying only 2 fibers. The Newton search reached the lowest number of correct fiber identifications producing numerous spurious results in all voxels.

FIGURE 5.

ODF-FP and the probabilistic approach were visibly more robust than other tested algorithms when reconstructing the number of fibers per voxel in the synthetic data experiment. The plots show percent rates of correctly identified voxels with N = 1, 2, 3 crossing fibers (respectively).

When comparing ODF-FP with FSL’s Bedpostx, we must emphasize the significant differences in runtimes of the above. The processing of 100000 voxels on a Dell^® laptop with Intel^® Core™ i7-9850H CPU @ 2.60GHz and 32GB RAM took respectively: 3h20min. (Probabilistic), 30min. (ODF-FP), 25sec. (CSD), 14sec. (Newton search), and 8sec. (Local max). The ODF-dictionary generation time varied depending on the dictionary size: 400sec. (for n = 10⁶), 38sec. (n = 10⁵), 6sec. (n = 10⁴), 2sec. (n = 10³). The most time-consuming part of ODF-FP was dedicated to finding a dominant peak inside each data voxel and calculating the rotation matrices. Nonetheless, Probabilistic remained an order of magnitude more complex.

As we mentioned earlier, our ODF-FP approach also allowed for looking up microstructure parameters of the assumed biophysical model. Despite the large number of DOF (16 in the full model with N = 2 fibers, and 9–12 in the simplified ones), ODF-FP produced more accurate estimations of the intra- and extra-axonal diffusivities than MIX Optimizer and the Brute2Fine algorithms (Figure 6). The root mean square errors and standard deviations (RMSE ± STD) were visibly lower in almost all studied ODF-dictionary variants (except for extra only) and ODF-dictionary sizes than in MIX optimizer and Brute2Fine. Notably, the two latter algorithms gave results near random guess, i.e. RMSE ± STD ≈ 0.5±0.5. Also note that the superior performance of ODF-FP was even more pronounced when reconstructing the intra-axonal fraction sizes f_in (Figure 6). More detailed analysis (Figure S1) showed that ODF-FP tended to select the microstructure parameters near the center of the plausible ranges, while MIX optimizer and Brute2Fine were biased towards the lower bounds of the ranges.

FIGURE 6.

Results of fitting of the diffusivity parameters D_a,∥, D_e,∥, D_e,⊥ and the intra-axonal volume fraction f_in with ODF-FP in the synthetic data experiment. The plots present normalized root mean squared errors with standard deviations (RMSE±STD) when using the full model (a) and its three simplifications (b-d). The ODF-dictionaries contained 10³, 10⁴, 10⁵, or 10⁶ elements (respectively). The results are compared to the state-of-the-art fitting algorithms: MIX optimizer and Brute2Fine.

In vivo diffusion data results

The diffusion data downsampled to 2.5 mm isotropic resolution posed an easier task for the tractography algorithm than those at 1.25 mm due to higher SNR and lower level of detail. In this vein, we compared the shapes of PyT, AF, and OR reconstructed at the high and low resolution. The largest differences were visible when dissecting PyT (Figure 7). Local max and Newton search were unable to reconstruct the fanning of the fibers. The outcomes of CSD, Probabilistic, and ODF-FP displayed the most realistic shapes. Nonetheless, the tracts produced with CSD and Probabilistic slightly deteriorated at the high resolution, while our proposed approach performed equally well at both levels of detail. The shapes of AF and OR were generally similar for all techniques, although ODF-FP, CSD, and the probabilistic method reproduced more cortical terminations of the tracts (Figure 7).

FIGURE 7.

Cortical terminations of the fascicles were most abundantly reconstructed with ODF-FP, and slightly less with the probabilistic approach or CSD. The images show deterministic tractography reconstructions of the pyramidal tracts, arcuate fasciculus, and optic radiations virtually dissected in DSI Studio from an HCP subject in the original (1.25 mm isotropic) and reduced (2.5 mm isotropic) resolution, using 5 alternative techniques for determining fiber numbers and directions (presented in columns).

In the quantitative experiment, ODF-FP, CSD, and Probabilistic ensured a similar balance between the true and false results (Figure 8a). However, CSD produced slightly less true positives than ODF-FP, while Probabilistic found more false positives than our approach. On the contrary, Local max and Newton search hardly identified more than 1 fiber per voxel, which resulted in relatively few true positives and almost no false positives, yet many false negatives.

FIGURE 8.

The low-resolution reconstructions of fibers of an HCP subject using CSD, the probabilistic approach, and ODF-FP reached highest agreement with the high-resolution reference data of the same subject (top-left slice) processed in DSI Studio. The first row of (a) presents the number of fibers per voxel as found by the tested approaches. The fibers reconstructed in the low-resolution data whose orientation matched the fibers in the respective high-resolution voxels are classified as true positives (TP; second row). The last two rows of (a) illustrate, respectively, the false positives (FP) and false negatives (FN) of this comparison. The relative TP and FP rates are plotted in (b) and (c)

The TP rate, defined in Equation 11, oscillated around 50% in ODF-FP, CSD, and Probabilistic. The ODF-FP with the dictionary generated from the full model reached the highest score (56%). Most variants with the simplified or degenerated ODF-dictionaries were comparable (52 – 56%) except for the intra only and extra only (both 47%). All the methods based on ODF peak finding had lower rates ranging between 41% and 50% (Figure 8b).

Local max and Newton search scored seemingly best FP rates (Figure 8c), although these outstanding results came with little TPs and many FNs (Figure 8a). CSD and our approach reached below 20% of FP rates in almost all tested variants (except intra only), outperforming Probabilistic.

Finally, let us note that the diffusivities obtained with ODF-FP on the HCP data conformed with the values reported in literature [27, 28, 29]. In corpus callosum, most of our reconstructed intra-axonal diffusivities D_a,∥ ranged between 2.3 and 2.5 · 10⁻⁹ m²/s, while the parallel extra-axonal diffusivities D_a,∥ typically lied in 1.9–2.4 · 10⁻⁹ m²/s (Figure 9). The perpendicular extra-axonal diffusivities D_e,⊥ were less than a half of D_e,∥, conforming to the extra-axonal space tortuosity levels reported in other studies [21, 42, 57, 58]. We also observed higher intra-axonal volumes f_in and fiber fractions p⁽¹⁾ in corpus callosum than in the ensemble of WM. Finally, our predominantly low values of f_in agreed with the known dominance of extra-axonal diffusion inside brain tissue [59, 60, 61]. Figure 9 shows a representative set of microstructure parameters related to the dominating fiber and the free water fraction of the full model. In the supporting information figures, we provide the summary of the remaining parameters of the full model (Figures S2 and S3) and the results obtained with the equal fibers model (Figures S4 and S5). In this experiment, we did not consider any further simplifications of the diffusion model, since they disregarded the intra- or extra-axonal contribution.

FIGURE 9.

Diffusivities and fraction volumes found with our method in vivo conform with the values reported in the literature. The microstructure parameters representing the dominating fiber of the full model $D_{a,\parallel }^{(1)}$, $D_{e,\parallel }^{(1)}$, $D_{e,\perp }^{(1)}$, $f_{in}^{(1)}$,p⁽¹⁾ and the free water fraction p_iso are given in rows. The second column from the left presents color maps plotted at a sample axial slice with color intensities determined by the respective intervals. The remaining three columns on the right show histograms calculated in the regions of genu and splenium of corpus callosum (CC) and the ensemble of WM. The diffusion-weighted images were provided by the Human Connectome Project.

5 ∣. DISCUSSION

In this work, we applied ODF-FP with the biophysical multicompartment diffusion model to reconstruct WM fibers crossing at shallow angles which are challenging for contemporary techniques. Our approach — based on matching ODF-fingerprints calculated from the input signal and synthetically generated ODF-fingerprints stored in a dictionary — outperformed the crossing angles reconstruction accuracy of Probabilistic, CSD, and local maximum search methods (Figures 3 and 5), contributing to more accurate tractography of major fascicles (Figure 7).

In addition, we showed that ODF-FP is able to fit the microstructure parameters of our model without the need to explore a high-dimensional space of parameters. Our method was more accurate than the two state-of-the-art fitting mechanisms — MIX optimizer and Brute2Fine — in the experiment with synthetic data (Figure 6). In vivo, the diffusivity values and fraction volumes found with our method agreed with the values reported in other studies (Figure 8).

It is also worth mentioning that our proposed randomization of the ODF-dictionary generation mechanism replaced the memory-consuming enumerated approach used earlier in ODF-FP (Figure 2). In result, we were able to construct ODF-dictionaries of arbitrary sizes, using multidimensional diffusion models.

5.1 ∣. High-accuracy reconstruction of shallow crossing angles

Reconstruction of shallow crossing angles is inherently limited [8, 14, 36, 54, 62, 63, 64]. In a typical scenario, fibers crossing at angles below 40° are barely distinguishable from a single fiber regardless of the angular precision of acquisition [6, 7, 14, 62, 65, 66]. The advent of CSD [8, 14, 54] helped reduce blurring artifacts in ODFs, although the technique itself remained relatively noise-sensitive and prone to spurious fiber identification.

In this work, we aimed to break the shallow angle barrier using ODF-FP with the biophysical multicompartment model. Our synthetic data experiment showed that ODF-FP was able to identify fibers crossing at angles as shallow as 10°. Certainly, the angular precision deteriorated below 40°, especially under low SNR, yet still ODF-FP largely outperformed the other tested approaches. In particular, our method successfully identified crossing fibers in many cases where ODF peak finding techniques wrongly recognized them as a single fiber. Such a misclassification observed in the latter approach led to disruption in the tractography results (Figure 7). Simultaneously, a part of the improvement in the ODF-FP tractography results observed at the cortical terminations of PyT, AF, and OT may also be attributed to the use of a voxel-dependent biophysical model that better captures the organization of cortical gray matter than WM specific models [45].

5.2 ∣. ODF-FP as a parameter fitting mechanism

As any other method for fitting the microstructure parameters, ODF-FP is susceptible to local optima. This is inevitable, especially for the diffusion models with many DOF [21, 42]. Similarly to Daducci et al. [24], we perceive our approach as a relaxation of the high-dimensional problem of model fitting with the linear problem of pattern matching. The parameter space is too large for contemporary personal computers to fully explore. ODF-FP offers a trade-off between the problem complexity and the accuracy of the solution.

5.3 ∣. Recommendations for ODF-dictionary generation

The use of ODF-FP comes with numerous design decisions concerning the variant of the diffusion model, the ODF-dictionary size, and the distribution of synthetically generated ODF-dictionary elements, each of which may impact the outcome. Our study of fibers crossing at shallow angles allowed us to formulate some guidance that we provide below.

Variant of the diffusion model

Our initial multicompartment model, which we dubbed full model, assumed a full variability of diffusivities and fraction sizes. It proved accurate in terms of angular precision of crossing fibers and was relatively robust to noise. Also, the microstructure parameters reconstructed in vivo were in agreement with other studies. However, such a flexible model is expected to have considerable plateau areas around global optima, resulting in a broad set of optimal or suboptimal solutions [21, 42]. It is then reasonable to incorporate a set of constraints of the diffusion model to guide the fitting process. For instance, the well-established NODDI model [43] assumes constant parallel diffusivities and a fixed tortuosity level, while CHARMED [67] considers a single extra-axonal compartment.

With ODF-FP, we managed to reduce the number of DOF by adding the equal fibers assumption without visibly sacrificing the accuracy of reconstruction. Most of our further simplifications also had a minor impact on the angular precision of crossing fibers reconstruction. One notable exception was the extra only simplification, which essentially boiled down to the multi-tensor model. As it turned out, extra only was the least effective among all the tested ODF-FP variants, supporting our argument that the multicompartment diffusion model significantly improves the performance of ODF-FP.

ODF-dictionary size

All our dictionary sizes — spanning from 10³ to 10⁶ — gave comparably good results when we used the diffusion model with 2 crossing fibers per voxel. Neither the angular accuracy, nor the reconstruction of diffusivities have notably changed as we shifted the ODF-dictionary size by 3 orders of magnitude. In most of our experiments with N ≤ 2, we chose to use ODF-dictionaries having 10⁵ elements. This was the maximum size for which we could solve the ODF-FP matching problem defined in Equation 4 without splitting the input data into chunks.

In the more general variant with N ≤ 3 crossing fibers per voxel, we used 10⁶ elements instead. Theoretically, our ODF-dictionary could have covered all directions determined by the sphere tessellation with fewer samples (cf. Table 1). In practice, this would require a deterministic control mechanism to guarantee that each direction was selected at least once. We preferred to avoid such manual manipulation and kept the stochastic model of the ODF-dictionary. We thus chose larger dictionary sizes to ensure sufficient coverage of the parameter space

Distribution of ODF-dictionary elements

We are aware that many samples in the uniformly distributed ODF-dictionary may never match the input. To mitigate this, one could use the normal distribution centered at some “expected values” instead. Such a modification could speed up the ODF-FP run time and possibly ensure uniqueness of the diffusivity parameters. However, it would also introduce a strong bias towards assumed priors. We believe that such a bias should be avoided, even at the cost of data redundancy. Notably, the uniform distribution of ODF-dictionary elements seems better suited for clinical images containing lesions or other anomalies that might be missed or transformed otherwise. Future work should address this issue.

5.4 ∣. Limitations of the method

The ODF-FP requires a properly designed ODF-dictionary. We showed that our randomized generation mechanism is resistant to certain mismatched parameter ranges, although the accuracy of our approach still mostly relies on the generated ODF-dictionary elements. In particular, noisy or otherwise corrupted DWIs are challenging for ODF-FP, as they may result in producing spurious fibers. The parameter ranges of the diffusion model should thus be chosen carefully to avoid overfitting.

In this paper, we provided a set of good practices to generate an ODF-dictionary for reconstruction of fibers crossing at narrow angles. Despite that, a period of trial and error might still be necessary to tune the dictionary to particular needs.

Another issue is the potential impact of the MRI acquisition on efficacy of ODF-FP. In our study, we used densely sampled DWIs acquired at multiple b-shells. Future work should address the optimization of the imaging protocol aimed at transferring our proposed approach to the requirements and limitations of clinical practice.

6 ∣. CONCLUSION

We studied the ability of ODF-FP equipped with the biophysical multicompartment diffusion model to identify fibers crossing at shallow angles. Our approach outperformed the commonly used methods based on ODF peak finding in both synthetic and in vivo diffusion data. This helped us obtain more plausible tractography outcomes, particularly at the challenging cortical terminations of the tracts.

In addition, we showed the capability of our approach to find the diffusivity parameters and the fraction sizes more effectively than the state-of-the-art algorithms do. By replacing the high-dimensional parameter space exploration with the ODF-FP matching problem, we managed to reduce the computation time and memory size needed to fit the multicompartment diffusion model.

Supplementary Material

1

SUPPORTING INFORMATION FIGURE S1 Accuracy of the microstructure parameters reconstruction in the in silico experiment. The plots show a representative example of the dominating fiber in the full model. The ground truth values are grouped into intervals illustrating the accuracy inside the plausible ranges of parameters.

SUPPORTING INFORMATION FIGURE S2 Diffusivities and fraction volumes reconstructed in vivo using ODF-Fingerprinting with the full biophysical multicompartment diffusion model. The plots show the second fiber and the free water fraction.

SUPPORTING INFORMATION FIGURE S3 Diffusivities and fraction volumes reconstructed in vivo using ODF-Fingerprinting with the full biophysical multicompartment diffusion model. The plots show the third fiber and the free water fraction.

SUPPORTING INFORMATION FIGURE S4 Diffusivities D_a,∥, D_e,∥, D_a,⊥ and intra-axonal fraction volumes f_in reconstructed in vivo using ODF-Fingerprinting with the biophysical multicompartment diffusion model under the equal fibers assumption.

SUPPORTING INFORMATION FIGURE S5 Freewater p_iso and fiber fraction volumes p⁽¹⁾, p⁽²⁾, p⁽³⁾ reconstructed in vivo using ODF-Fingerprinting with the biophysical multicompartment diffusion model under the equal fibers assumption.

Data availability statement

The Matlab^®-based source code of ODF-FP used in this work is publicly available at https://bitbucket.org/sbaete/odffingerprinting

Alternatively, the Python version of ODF-FP implemented as an extension of the DIPY library can be downloaded from https://github.com/filipp02/dipy_odffp/tree/odffp