#
Reference Tracts and Generative Models for Brain White Matter Tractography^{ †}

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Participants

#### 2.1.1. Training Data

#### 2.1.2. Testing Data

#### 2.2. MRI

^{2}) and sets of diffusion-weighted (b = 1000 s/mm

^{2}) single-shot, spin-echo, echo-planar (EP) imaging volumes, acquired with diffusion gradients applied in 64 non-collinear directions [22] and 2 mm isotropic spatial resolution.

#### 2.3. Image Analysis

#### 2.4. Reference Tracts

#### 2.4.1. Atlas-Based Reference Tracts

#### 2.4.2. Data-Based Reference Tracts

#### 2.5. Creation of Matching Models

#### 2.6. Testing of Reference Tracts and Matching Models

#### 2.7. Sampling from PNT Models

- Identify the image voxel corresponding to the reference anchor point, and choose a specific starting location from a uniform distribution over that voxel. Note this as the first pseudo-knot point.
- Sample ${L}_{1}$ and ${L}_{2}$ from their respective distributions, thereby obtaining the length of the sample streamline either side of the anchor point.
- Beginning at the point obtained in step 1, sample ${\mathbf{v}}_{u}$ sequentially for u $\in $ {−1, ..., $-{L}_{1}$}. In each case, take a step of length d in the direction of ${\mathbf{v}}_{u}$ from the current pseudo-knot point to arrive at the next pseudo-knot point.
- Return to the point obtained in step 1, and sample ${\mathbf{v}}_{u}$ sequentially for u $\in $ {1, ..., ${L}_{2}$}, analogously to step 3.
- Use B-spline interpolation to recover a curve between the sequence of pseudo-knot points.

- Sample ${\varphi}_{u}$ from the model.
- Establish a point,
**w**, on the plane passing through the origin perpendicular to ${\mathbf{v}}_{u}^{*}$. The equation of this plane is ${\mathbf{v}}_{u}^{*}\xb7\mathbf{w}=0$, so any vector perpendicular to ${\mathbf{v}}_{u}^{*}$ will do. We take $\mathbf{w}={\mathbf{v}}_{u}^{*}\times \widehat{\mathbf{x}}$, where $\widehat{\mathbf{x}}$ = (0, 0, 1) unless this is collinear with ${\mathbf{v}}_{u}^{*}$, in which case we use $\widehat{\mathbf{x}}$ = (1, 0, 0). - Sample θ ~ $\mathcal{U}$(0, 2π), the angle around the locus circle.
- Rotate
**w**by the angle θ around the unit vector ${\widehat{\mathbf{v}}}_{\mathit{u}}^{*}={\mathbf{v}}_{u}^{*}/\Vert {\mathbf{v}}_{u}^{*}\Vert ={\mathbf{v}}_{u}^{*}/d$, using Rodrigues’ rotation Formula (1):$${\mathbf{w}}^{\prime}=\mathbf{w}\mathrm{cos}\theta +{\widehat{\mathbf{v}}}_{\mathit{u}}^{*}({\widehat{\mathbf{v}}}_{\mathit{u}}^{*}\xb7\mathbf{w})(1-\mathrm{cos}\theta )+(\mathbf{w}\times {\widehat{\mathbf{v}}}_{\mathit{u}}^{*})\mathrm{sin}\theta $$ - Scale
**w**′ to the radius of the locus circle and translate it along the reference vector, to arrive at the final step vector, ${\mathbf{v}}_{u}$, as (2):$${\mathbf{v}}_{u}=\frac{{\mathbf{w}}^{\prime}d\mathrm{sin}{\varphi}_{u}}{\Vert {\mathbf{w}}^{\prime}\Vert}+{\mathbf{v}}_{u}^{*}\mathrm{cos}{\varphi}_{u}$$

#### 2.8. Creating Synthetic Tracts from PNT Models

## 3. Results

#### 3.1. Testing of Reference Tracts and Matching Models

#### 3.1.1. Visual Assessments

#### 3.1.2. FA and MD Variability

#### 3.1.3. Overlap Analysis

#### 3.2. Assessment of Synthetic Tracts Sampled from PNT Models

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Flow chart of the processes followed in this manuscript. Black paths show the creation of the data-based reference tracts and training data-based supervised models (which represent the deviations of the training data), using the training data. Color paths show the three cases of tract segmentation performed in the LBC1936 data: red paths use the data-based reference tracts and the training data-based models to segment white matter tracts; blue paths use the data-based reference tracts in the LBC1936 data to create models (which represent the deviations of the tracts corresponding to LBC1936 data), and segment the tracts simultaneously using expectation–maximization (EM); and, yellow paths use the atlas-based reference tracts to create models (which represent the deviations of the tracts corresponding to LBC1936 data), and segment the tracts simultaneously using EM.

**Figure 2.**Graphical representation of a candidate tract (

**a**), the median line is fitted to a B-spline with knot points separated by a distance d (straight-line distance). A B-spline representation is also used for the reference tract. The vector between two consecutive knot points in the candidate and the equivalent knot points in the reference can be compared and the angular deviations obtained. (

**b**) illustrates the shape model used by probabilistic neighborhood tractography (PNT), based on angular deviations ${\varphi}_{u}$, between equivalent tract segments in the reference and candidate tracts, ${\mathbf{v}}_{u}^{*}$ and ${\mathbf{v}}_{u}$, respectively. The putative direction of each segment is always away from the anchor point. Adapted from [5,6].

**Figure 3.**Graphical representation of the sampling process for step vectors, ${\mathbf{v}}_{u}$. (

**a**) From the voxel corresponding to the anchor point, the “left” and “right” tract lengths are sampled from the model length distributions, obtaining the total length of the streamline. (

**b**) From the first step on one side, the vector ${\mathbf{v}}_{u}$ is sampled, leading to the next knot in the streamline. This vector is obtained from the angle ${\varphi}_{u}$ sampled from the model angle distribution at that knot. This is replicated for every step until the distance ${L}_{2}$ is reached. The process is then repeated for the “left” tract lengths. (

**c**,

**d**) Geometric representation of the sub-steps for the sampling of ${\mathbf{v}}_{u}$: given a reference tract direction, ${\mathbf{v}}_{u}^{*}$, and an angular deviation from it, ${\varphi}_{u}$ (

**c**). These jointly specify a circular locus of possible directions (

**d**), from which a final vector is chosen by additionally sampling θ $\in $ [0, 2π].

**Figure 4.**Group maps projections for the 16 tracts of interest segmented using the data-based (left panel) and atlas-based (right panel) reference tracts. Top panels used a matching model trained in the LBC1936 data, and the bottom panel used a model trained in the training data. The tracts represented are: (

**a**) genu and (

**b**) splenium of the corpus callosum, left and right arcuate fasciculus (

**c**,

**d**), left and right anterior thalamic radiation (

**e**,

**f**), left and right inferior longitudinal fasciculus (

**g**,

**h**), left and right dorsal (

**i**,

**j**) and ventral (

**k**,

**l**) cingulum, left and right corticospinal tracts (

**m**,

**n**) and left and right uncinate fasciculus (

**o**,

**p**). Color scale represents the voxel visitation frequency, from 1 (light yellow) to 50 (dark blue). Maps are projected into the plane of the voxel with maximum visitation value. Red arrows point at the main differences obtained between the resulting tracts derived from atlas-based and data-based reference tracts. Figure adapted from [1].

**Figure 5.**Overlays of the uncinate (

**a**) and arcuate (

**b**) fasciculi. Atlas tracts represented in red (from [17]) and tracts segmented in the LBC1936 data using atlas-based reference tracts and unsupervised models in green (

**left**) and blue (

**right**), in radiological convention.

**Figure 6.**Streamline representations of the synthetic tracts obtained by sampling from the PNT models generated from the training and LBC1936 data. First column: PNT model from the training dataset using the data-based reference tract; second column: PNT model from the LBC1936 dataset using the data-based reference tract; third column: PNT model from the LBC1936 dataset and the atlas-based reference tract. (

**a**) genu (

**b**) splenium, (

**c**) Arc, (

**d**) ATR, (

**e**) Cing, (

**f**) Cing, ventral, (

**g**) ILF, (

**h**) Unc, and (

**i**) CST.

**Table 1.**Proportion of segmented tracts visually acceptable when using two different matching models and each set of reference tracts as priors.

Reference Tracts | Data-Based | Atlas-Based | |
---|---|---|---|

Model Trained on | Training Data | LBC1936 Data | LBC1936 Data |

Genu | 100.0% | 100.0% | 96.0% |

Splenium | 98.0% | 96.0% | 98.0% |

L Arc | 100.0% | 100.0% | 98.0% |

R Arc | 96.0% | 96.0% | 94.0% |

L ATR | 100.0% | 100.0% | 32.0% |

R ATR | 96.0% | 100.0% | 76.0% |

L ILF | 100.0% | 100.0% | 100.0% |

R ILF | 100.0% | 100.0% | 100.0% |

L Cing | 98.0% | 98.0% | 100.0% |

R Cing | 98.0% | 92.0% | 98.0% |

L Cing, ventral | 98.0% | 100.0% | 98.0% |

R Cing, ventral | 94.0% | 98.0% | 100.0% |

L CST | 100.0% | 98.0% | 100.0% |

R CST | 100.0% | 100.0% | 100.0% |

L Unc | 96.0% | 92.0% | 88.0% |

R Unc | 100.0% | 100.0% | 100.0% |

Mean | 98.3% | 98.1% | 92.4% |

**Table 2.**Averaged values of fractional anisotropy (FA) and mean diffusivity (MD) measured along the tracts segmented with two different matching models, and atlas-based or data-based reference tracts as priors in 50 older age volunteers (LBC1936).

FA | MD (10^{−6} mm^{2}/s) | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Reference | Atlas-Based | Data-Based | Atlas-Based | Data-Based | ||||||||||||||

Model Training | LBC1936 Data | LBC1936 Data | Training Data | LBC1936 Data | LBC1936 Data | Training Data | ||||||||||||

Mean (sd) | CV | Mean (sd) | CV | Mean (sd) | CV | Mean (sd) | CV | Mean (sd) | CV | Mean (sd) | CV | |||||||

Genu | 0.41 | (0.05) | 0.11 | 0.39 | (0.05) | 0.12 | 0.39 | (0.05) | 0.12 | 776.91 | (65.59) | 0.08 | 799.20 | (75.46) | 0.09 | 799.85 | (74.59) | 0.09 |

Splenium | 0.45 * | (0.09) | 0.20 | 0.52 * | (0.06) | 0.12 | 0.51 * | (0.08) | 0.15 | 1117.26 * | (220.22) | 0.20 | 807.61 * | (108.59) | 0.13 | 837.77 * | (162.71) | 0.19 |

L Arc | 0.46 | (0.05) | 0.10 | 0.45 | (0.04) | 0.09 | 0.45 | (0.04) | 0.10 | 663.30 | (49.21) | 0.07 | 661.30 | (49.26) | 0.07 | 659.82 | (49.73) | 0.08 |

R Arc | 0.43 | (0.05) | 0.12 | 0.42 | (0.04) | 0.10 | 0.43 | (0.04) | 0.09 | 646.56 | (55.00) | 0.09 | 645.36 | (48.93) | 0.08 | 644.13 | (45.30) | 0.07 |

L ATR | 0.34 | (0.05) | 0.14 | 0.34 | (0.03) | 0.10 | 0.34 | (0.03) | 0.10 | 757.89 | (81.23) | 0.11 | 755.39 | (60.94) | 0.08 | 746.41 | (60.30) | 0.08 |

R ATR | 0.35 * | (0.04) | 0.10 | 0.36 * | (0.03) | 0.08 | 0.33 * | (0.04) | 0.12 | 747.07 * | (54.08) | 0.07 | 704.05 * | (50.40) | 0.07 | 766.81 * | (74.85) | 0.10 |

L ILF | 0.42 | (0.05) | 0.12 | 0.41 | (0.05) | 0.12 | 0.40 | (0.05) | 0.12 | 740.50 | (75.45) | 0.10 | 752.41 | (67.06) | 0.09 | 745.86 | (61.13) | 0.08 |

R ILF | 0.39 | (0.05) | 0.14 | 0.40 | (0.04) | 0.11 | 0.38 | (0.05) | 0.12 | 788.00 | (142.54) | 0.18 | 750.31 | (83.70) | 0.11 | 755.39 | (87.47) | 0.12 |

L Cing | 0.45 | (0.05) | 0.12 | 0.46 | (0.06) | 0.12 | 0.46 | (0.06) | 0.12 | 647.29 | (51.00) | 0.08 | 638.39 | (45.15) | 0.07 | 640.95 | (47.46) | 0.07 |

R Cing | 0.42 | (0.06) | 0.13 | 0.43 | (0.04) | 0.10 | 0.42 | (0.05) | 0.11 | 619.92 | (36.16) | 0.06 | 626.56 | (36.03) | 0.06 | 630.97 | (33.82) | 0.05 |

L Cing, ventral | 0.32 | (0.06) | 0.19 | 0.29 | (0.04) | 0.12 | 0.29 | (0.04) | 0.12 | 752.54 | (155.54) | 0.21 | 728.86 | (62.50) | 0.09 | 733.07 | (69.52) | 0.09 |

R Cing, ventral | 0.30 | (0.06) | 0.20 | 0.30 | (0.05) | 0.15 | 0.29 | (0.04) | 0.14 | 760.68 | (95.07) | 0.12 | 748.37 | (79.00) | 0.11 | 748.73 | (88.67) | 0.12 |

L CST | 0.48 | (0.03) | 0.07 | 0.46 | (0.04) | 0.08 | 0.46 | (0.04) | 0.08 | 655.47 | (36.72) | 0.06 | 672.26 | (37.18) | 0.06 | 675.52 | (38.65) | 0.06 |

R CST | 0.49 | (0.03) | 0.07 | 0.49 | (0.03) | 0.07 | 0.50 | (0.04) | 0.07 | 653.82 * | (32.72) | 0.05 | 676.03 * | (32.36) | 0.05 | 676.37 * | (31.99) | 0.05 |

L Unc | 0.34 | (0.03) | 0.10 | 0.33 | (0.03) | 0.10 | 0.34 | (0.04) | 0.11 | 767.04 | (53.54) | 0.07 | 767.63 | (60.41) | 0.08 | 764.88 | (60.65) | 0.08 |

R Unc | 0.33 | (0.03) | 0.10 | 0.33 | (0.03) | 0.10 | 0.33 | (0.04) | 0.11 | 756.22 | (41.27) | 0.05 | 758.75 | (41.27) | 0.05 | 754.75 | (41.77) | 0.06 |

Mean | 0.40 | (0.06) | 0.13 | 0.40 | (0.07) | 0.10 | 0.40 | (0.07) | 0.11 | 740.65 | (115.51) | 0.10 | 718.28 | (58.64) | 0.08 | 723.83 | (61.36) | 0.09 |

Reference Tracts | Data-Based | Atlas-Based | |
---|---|---|---|

Model Trained on | Training Data | LBC1936 Data | LBC1936 Data |

Genu | 0.46 | 0.50 | 0.43 |

Splenium | 0.63 | 0.62 | 0.48 |

L Arc | 0.34 | 0.34 | 0.21 |

R Arc | 0.36 | 0.34 | 0.22 |

L ATR | 0.31 | 0.31 | 0.28 |

R ATR | 0.37 | 0.34 | 0.35 |

L ILF | 0.44 | 0.43 | 0.4 |

R ILF | 0.50 | 0.49 | 0.48 |

L Cing | 0.52 | 0.49 | 0.57 |

R Cing | 0.52 | 0.51 | 0.58 |

L Cing, ventral | 0.47 | 0.48 | 0.45 |

R Cing, ventral | 0.49 | 0.49 | 0.47 |

L CST | 0.52 | 0.52 | 0.52 |

R CST | 0.65 | 0.63 | 0.57 |

L Unc | 0.27 | 0.26 | 0.22 |

R Unc | 0.29 | 0.28 | 0.29 |

Range | 0.27–0.65 | 0.26–0.63 | 0.21–0.58 |

Mean | 0.45 | 0.44 | 0.41 |

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**MDPI and ACS Style**

Muñoz Maniega, S.; Bastin, M.E.; Deary, I.J.; Wardlaw, J.M.; Clayden, J.D. Reference Tracts and Generative Models for Brain White Matter Tractography. *J. Imaging* **2018**, *4*, 8.
https://doi.org/10.3390/jimaging4010008

**AMA Style**

Muñoz Maniega S, Bastin ME, Deary IJ, Wardlaw JM, Clayden JD. Reference Tracts and Generative Models for Brain White Matter Tractography. *Journal of Imaging*. 2018; 4(1):8.
https://doi.org/10.3390/jimaging4010008

**Chicago/Turabian Style**

Muñoz Maniega, Susana, Mark E. Bastin, Ian J. Deary, Joanna M. Wardlaw, and Jonathan D. Clayden. 2018. "Reference Tracts and Generative Models for Brain White Matter Tractography" *Journal of Imaging* 4, no. 1: 8.
https://doi.org/10.3390/jimaging4010008