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Edited by: Xiaoying Tang, SYSU-CMU Joint Institute of Engineering, China

Reviewed by: Suyash P. Awate, Indian Institute of Technology Bombay, India; Chuyang Ye, Institute of Automation (CAS), China

*Correspondence: Xiaoming Dong

This article was submitted to Brain Imaging Methods, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The problem of estimating neuronal fiber tracts connecting different brain regions is important for various types of brain studies, including understanding brain functionality and diagnosing cognitive impairments. The popular techniques for tractography are mostly sequential—tracts are grown sequentially following principal directions of local water diffusion profiles. Despite several advancements on this basic idea, the solutions easily get stuck in local solutions, and can't incorporate global shape information. We present a global approach where fiber tracts between regions of interest are initialized and updated via deformations based on gradients of a posterior energy. This energy has contributions from diffusion data, global shape models, and roughness penalty. The resulting tracts are relatively immune to issues such as tensor noise and fiber crossings, and achieve more interpretable tractography results. We demonstrate this framework using both simulated and real dMRI and HARDI data.

This paper considers an important problem of estimating major white matter fiber tracts in human brain using diffusion magnetic resonance imaging (dMRI) images (Mori et al., ^{2}) is estimated (Descoteaux,

Due to the importance of tract-based connectivity in brain connectomic analysis, there have been a number of solutions developed for estimating fiber tracts. They can be loosely grouped into two categories: local and global methods. Local methods construct fiber curves sequentially based on the estimated local diffusion directions. Depending on the mechanism for specifying a local propagation direction, one can further classify the local methods into deterministic methods (Mori et al.,

We can summarize the limitations of current methods as follows: (a) The local methods are essentially sequential—they start fibers from one end and grow them over time. This one-boundary solution is not natural for tractography, which is actually a two-boundary problem. (b) The local tractography algorithms are highly susceptible to fiber crossing, noise and imaging artifacts. Incorrect recording or noisy observations of tensors can send algorithms in wrong directions and it is difficult to recover from such misdirections. (c) The global tractography algorithms achieve better stability with respect to noise, but they are very computationally expensive. (d) Both local and global methods tend to produce a large proportion of false positive fibers because of the noise and ambiguity at fiber crossings. Figure

Two examples of the classic streamline method does not work. The blue lines are ground truth fibers. The red and green lines are the tractorgraphy results from the FACT method. Starting from area A, FACT failed to reconstruct the fiber tracts that connect A and B.

In this paper, we propose a new approach that is essentially a global method but using additional geometry information for ensuring optimal solutions. The proposed method is fast and easy to implement, and robust to the noise in the data. Most importantly, it can incorporate the prior knowledge from anatomical structure and brain connectomics. Rather than growing fiber tracts sequentially, our idea is to initialize fiber tracts between regions of interest as Euclidean curves and then

In contrast to the probabilistic tractography method (Behrens et al.,

The rest of this paper is organized as follows. We describe the three components of the posterior energy—data likelihood, shape prior and roughness penalty—and their gradients in Section 2. The resulting tractography algorithm is laid out in Section 3, and experimental results using both simulated and real data, the extension to HARDI data are presented in Section 4. We close the paper with a short discussion in Section 5.

Although the framework can be easily generalized to 3D data, we will restrict to 2D data in this paper for simplicity. The theory is general enough to be applicable to 3D data directly.

First, we develop a mathematical framework for estimation of fiber tracts using tensor data and prior shape models. Let ^{2}, let

This total energy functional has contributions from three different criteria that are weighted by the coefficients λ_{1}, λ_{2}, λ_{3} > 0. The data energy _{data} is defined solely from the diffusion data in the image, _{prior} is the prior shape energy defined from a statistical model on shapes of the fiber β, and the smoothing energy _{smooth} is a penalty that ensures a certain amount of smoothness in the estimated fiber. In order to minimize E_{total} we use a gradient descent procedure that updates the curve according to β ↦ β − δ∇_{β}

That is, we search for a local minimization of Equation (1) via gradient descent. The weights λ_{i} will certainly affect curve evolution, i.e., a large penalty on the smoothness term favors shorter fibers and so on. Through trial and error, one can adjust the λ's depending on the data and problem context. In the next three sections, we summarize the formulation of each of the three energy terms.

The data term is designed to quantify the agreement of the fiber directions with the diffusion tensor at that location. Let

_{data}, where black is the initial curve and red is the final curve. _{data} during this optimization.

Here _{β}(_{β(t)} is the tensor at location β(

We motivate the choice of this expression by focusing on some Riemannian frameworks used in tractography:

_{M} are the longest paths between given points in

_{β}(_{β(t)}, and not on the speed of traversal at β(_{data}[β] ≠ _{data}[β ◦ γ]. If that invariance is desired, one can achieve it by changing the measure of integration from

The next step is to derive the gradient of _{data} with respect to β for use in gradient-based optimization. To specify the gradient of _{data}, we need some additional notation. Note that for any location _{1}, _{2}) ∈ _{x1}_{x}, ∇_{x2}_{x} ∈ _{Mx}(_{x}_{x} is a higher-order tensor of the size 2 × 2 × 2. For any such tensor ^{2 × 2 × 2} and a vector ^{2}, we will use the notation: 〈〈_{1}_{2}_{M(x)}(_{data} as follows.

_{data} with respect to^{2}

A derivation of this expression is presented in the _{β}_{data} makes the optimization problem more efficient, as compared to purely numerical solutions.

Figure _{data} in the middle panel. It shows a tensor field _{β}_{data} and the result is drawn as a red curve. The corresponding evolution of _{data} is plotted in the right panel.

For regulating smoothness of the estimated curve, we follow a common approach from geometric active contours that is motivated in part by Euclidean heat flow. Define the smoothing energy function as _{smooth} is given by the Euclidean heat flow equation ∇_{smooth}(β) = κ_{β}_{β}, where κ_{β} is the curvature at each point of β and _{β} is the unit normal field along the curve. It is well known that this particular penalty on a curve's length leads to simultaneous smoothing and shrinking of a curve. If we rescale the curve to keep the original length, the main effect is that of smoothing. An example of this idea is illustrated in Figure _{smooth}. The left panel shows the initial curve (in black), and its updates using the negative gradient of _{smooth}. The corresponding decrease in _{smooth} is plotted on the right.

Evolution of a curve using negative gradient of _{smooth}. _{smooth}.

The next term to consider is _{prior} that forces the shapes of estimated fiber tracts to be similar to certain desired shapes. This term encodes the prior shape information about fibers connecting two ROIs, and is based on a statistical model that is learnt from the training or atlas data (generated by current local or global methods). In a brain connectome study framework, the brain is generally pre-segmented into small anatomical regions using software such as Freesurfer and ANTs (Avants et al., ^{2} metric under transformation. As a corollary, for any _{1}, _{2} ∈ 𝕃^{2}, we have ‖(_{1}, γ) − (_{2}, γ)‖ = ‖_{1} − _{2}‖, for any γ ∈ Γ, where Γ is the set of all orientation preserving diffeomorphisms of [0, 1]. Here (^{*}β, and the corresponding SRVF is given by ^{*}

Let β be a rescaled fiber curve such that it has unit length and let _{prior} in the active contour model is a function of β, but our statistical models are built on _{prior} can effectively encode the shape information and be invariant to the different sizes and coordinate systems of different brains. However, _{[μ]}(

Given a set of prior training shapes {[_{i}], _{m} be the _{m} to _{m} is:
_{m}, _{⊥} = _{m}_{‖}, _{m} is the diagonal matrix containing the first _{m}. Suppose now that we have a test shape [_{prior}(_{prior} with respect to _{prior} with respect to _{β}_{prior}(β) using a numerical approximation.

Figure _{prior}. The left panel shows the initial curve, and its updates using the negative gradient of _{prior}. The corresponding decrease in _{prior} is plotted on the right.

Evolution of a curve using negative gradient of _{prior}. _{prior}.

When we put together the three components of the energy, the shape of β is controlled by gradients of _{data}, _{prior} and _{smooth}, the smoothness is controlled by _{prior} and _{smooth}, and the nuisance variables (placement, scale, and rotation) are controlled only by _{data}. Now we summarize the overall algorithm for Bayesian tractography using the tensor field (Algorithm 1).

Bayesian Tractography Using Geometric Shape Priors

The advantage of the proposed framework is that it uses a global optimization to overcome issues such as fiber crossing and spatial noise. The final tracking result depends not only on the diffusion data, but also on prior shape information. The inclusion of shape prior distinguishes our method from other energy minimization based fiber-tracking algorithms, and is essential for the optimization procedure to come out of local solutions and reach a global solution. Most importantly, in our framework, the brain is parcellated into small regions, and the shapes of fibers connecting any pair of regions are found to be consistent. The proposed truncated wrapped-normal distribution can effectively capture the variation of shapes for each connection in the training data. In addition, since we reconstruct the whole fiber simultaneously by minimizing an energy function, the issue of fiber crossing has almost no detrimental effect of our fiber tracking algorithm.

As stated earlier, this Bayesian approach requires either a the training data or an atlas of fiber tracts between regions of interest, to estimate shape model and develop _{prior}. We can construct such data using existing tractography algorithms with maybe human inspection for quality control. However, since such a construction is needed only once, it can be performed offline.

In this section we present some results using both simulated and real data to illustrate the performance of the proposed method.

We first study our proposed tracking algorithm in the simulated settings. Let domain ^{2} for all our simulation examples. The tensor field on

In the experiment presented in Figure _{2} norm. We first calculate the distance of each fiber from the ground truth and then use the mean of all distances to quantify the difference between reconstructed fiber bundle and ground truth fiber bundle. The distances for each method are given in Figure

Tractography results on a simulated tensor field and distances from ground truth:

Additional details of this simulation experiment are presented in Figure _{total}. The left panel shows the initial curve (black), the final curve (red), and the ground truth curve (blue). The right panel shows the evolution _{total} during this iteration. In this experiment, we used the weights λ_{1} = 0.8, λ_{2} = 0.1, and λ_{3} = 0.1.

Detailed tractography results in the simulation example. Here we only focus on reconstruction of one of the curves. The black line is initialization, the red line is our result and the blue line is the ground truth.The right panel shows the evolution of the energy function.

Next, we apply our method to some real datasets—dMRI images downloaded from the Human Connectome Project (HCP) (Van Essen et al.,

An example of a sagittal slice of diffusion tensor data.

In the results presented here, we focus on estimating a set of fiber curves connecting the left and right superior frontal gyri. In order to generate a prior shape model, we use tracts extracted for 30 subjects between these regions as the training dataset. These tracts were manually identified with the help of Freesurfer Destrieux Atlas (Destrieux et al., _{i}] in the tangent space _{[μ]}(_{i}} in the vector space _{[μ]}(_{[μ]}(

Thirty training samples of fiber tracts, their Karcher mean and principal directions of shape variation. The rightmost panel from top to bottom represents the first 5 principal directions of variation in the training data.

Having developed a prior model for fiber shapes from the training data, we then apply our Bayesian method to the tensor data, especially focusing on the areas where the streamline method fails, and the results are presented in Figure _{data}, _{prior}, and _{smooth}—are shown in the middle row of this figure. Each one of these terms show a substantial decrease in their values during the iteration process.

Results comparison between streamline method and our method. In the

In order to study the impact of the weights λ_{1}, λ_{2}, and λ_{3} on the final result, we generated estimates for a few different combinations of these weights. The results are shown in the last row of this figure. In case where the weight for shape prior is high, the final result is close to the prior mean. In contrast, when the weight for the data term is high, there is a better agreement between the curve and the tensor field.

Another example of this Bayesian estimation is presented in Figure

Another example similar to Figure

The proposed framework can be extended to HARDI data, where an ODF is used to better represent the underlying diffusion profile. The data term is now defined as:

Here _{β}(_{p} is the ODF at _{data} with respect to β for use in gradient-based optimization. we can express the gradient of _{data} as follows.

_{data} with respect to^{2} _{total}_{total}

Tractography results on simulated ODF data. _{total}.

This paper introduces a Bayesian approach for estimating fiber tracts, between given pairs of points in a human brain, using dMRI and HARDI data. The basic idea is to define a composite energy functional, using a linear combinations of terms that relate to data, curve smoothness, and a prior shape model, and then use the gradient of this energy to iteratively optimize a contour. There are several novelties in this setup: (1) the data term is locally scale-invariant and measures only the agreement of the fiber direction with the given diffusion tensor field, (2) the length of the fiber is kept as a separate term, in order to have an additional control over fiber size, and (3) an external term involving statistical shape models, of fibers tracts connecting given regions, is used to improve optimization and interpretability. These shape models can come from training data developed using manual interventions or population atlases established from previous studies. The gradients of all the terms have analytical forms, making the gradient-based optimization very efficient. This framework is demonstrated successfully using simulated 2D tensor fields and 2D slices of volume dMRI data.

One advantage of our method is that it can naturally handle crossing bundles since we construct the streamline as a whole object. Relying on the prior shape information, we can reconstruct a fiber curve that have similar geometry to the prior knowledge. Figure

Examples showing that the proposed method can handle crossing and kissing fibers. Red lines are our tractograhy results, blue lines are ground truth and black lines are initializations. From upper left panel to bottom left panel, more and more crossing bundles are added into the simulation. The bottom right panel shows the shape prior used in our model.

However, the proposed Bayesian method needs to specify the starting and ending points for each extracted tract. To ensure that there is a tract between two ROIs, we currently rely on the atlas data. This procedure may end up with false positives, e.g., identifying a tract that does not exist. A future pruning procedure can be added as a post processing step, relying perhaps on the minimum energy as the reviewer has suggested. As another criterion, the diffusion profile along a tract can possibly be used as a feature to determine whether a tract is a false or a true positive.

As a future work, this framework can be naturally implemented using 3D dMRI data, and resulted tractography can be compared with some state of the art techniques.

All authors listed, have made substantial, direct and intellectual contribution to the work, and approved it for publication. XD, ZZ, and AS have contributed in development of theory and computer implementation.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research was supported in part by NSF grants DMS 1621787 and CCF 1617397 to AS. ZZ was partially supported by NSF grant DMS-1127914 to SAMSI. Data were provided in part by the HCP, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657).

In this section we derive an expression for the gradient of _{data}[β] with respect to β. Let ^{2} and _{data} in the direction of _{β}(

Here

Thus, the full gradient of _{data} with respect to β is given by:

Let's denote _{p} as the ODF at _{β(t)}(_{β}(_{data}[β] with respect to β. Let ^{2} and _{data} in the direction of

Here

where

Thus, the full gradient of _{data} with respect to β is given by: