Diffusion tensor magnetic resonance imaging (DTMRI) as a noninvasive modality providing in vivo anatomical information allows determination of fiber connections which leads to brain mapping. The success of DTMRI is very much algorithm dependent, and its verification is of great importance due to limited availability of a gold standard in the literature. In this study, unsupervised artificial neural network class, namely, self-organizing maps, is employed to discover the underlying fiber tracts. A common artificial diffusion tensor resource, named “phantom images for simulating tractography errors” (PISTE), is used for the accuracy verification and acceptability of the proposed approach. Four different tract geometries with varying SNRs and fractional anisotropy are investigated. The proposed method, SOFMAT, is able to define the predetermined fiber paths successfully with a standard deviation of (0.8–1.9) × 10−3 depending on the trajectory and the SNR value selected. The results illustrate the capability of SOFMAT to reconstruct complex fiber tract configurations. The ability of SOFMAT to detect fiber paths in low anisotropy regions, which physiologically may correspond to either grey matter or pathology (abnormality) and uncertainty areas in real data, is an advantage of the method for future studies.
Diffusion tensor magnetic resonance imaging (DTMRI, also called DTI) is a fundamental technique that allows in vivo structural brain imaging by white matter estimation [
An important drawback in the determination of the fiber paths for tractography purposes occurs in uncertainty regions where at least two fiber paths intersect. This study proposes an artificial neural network approach named SOFMAT based on self-organizing feature mapping (SOFM or SOM) to define the fiber tracts based on their diffusivity and to clarify, especially, the fiber tracts in these uncertainty regions [
The developed novel SOM-based tractography approach self-organizing feature mapping tractography (SOFMAT) is based on unsupervised learning method, which is used in the training of artificial neural networks (ANNs) [
Especially in studies dealing with complex data, ANN is very useful and preferable. The use of ANN has a wide range, such as analyzing seismic signals [
In our study, SOM is selected to train the ANN, because SOM as a classifier demonstrated successful identification of structured topologies in various domains [
The idea of SOFMAT is to accomplish the fiber pathways by considering each individual voxel’s contribution taking into account the neighboring voxels’ behavior in the topology. This is achieved by both the competing and the cooperating behavior of SOM nodes (neurons) in forming the topology. The proposed method has been tested on four phantom images from PISTE with various signal to noise (SNR) values. The images represent various levels of complexities involving crossovers, kisses, and direction changes. The results were then compared against well-accepted tractography algorithms reported in the literature (i.e., streamline (SLT) [
Preliminary studies indicate that SOFMAT method gives promising and relatively superior results compared to the traditionally implemented and well-accepted tractography algorithms mentioned above. SOFMAT has the ability to generate tracts in complex fiber structures such as the spiral phantom utilized and represented in this study. The main reason for the development of the SOFMAT method was to tract complex architectures like spiral trajectory where the standard streamline approaches were failing. A typical SLT algorithm follows only a single direction, where SOFMAT evaluates multiple directions regarding the topological neighborhood function [
The sections of the paper are organized as follows. In the next section, a brief background work related to the proposed method is introduced including the synthetic data resource utilized for evaluation. Section
The principles of DTI are based on the Stejskal-Tanner imaging sequence [ is the signal received with the
The diagonal and off-diagonal elements of
Principal component analysis (PCA) is used to perform the diffusion tensor analysis and compression. The diagonalization of the diffusion tensor as in (
In an attempt to identify nerve fiber trajectories, several DTI based tractography techniques have been proposed to propagate diffusion tensor fields. Since it is difficult to validate the findings of a tractography method on brain images, artificially produced validated phantom images are used for benchmarking. One such commonly utilized dataset in DT-MRI tractography literature is called “phantom images for simulating tractography errors” (PISTE). PISTE comprises a set of simulated fiber trajectories designed for testing, validating, and comparing tractography algorithms allowing the investigation of various geometries like linear, linear break, orthogonal crossing, and spiral [
The DT images are generated on the investigated trajectory with a decreasing anisotropy along the length of the tract, which is overlaid on a homogeneous anisotropic background. The data used in this study is available as 32 bit float binary files at
Self-organizing feature mapping tractography (SOFMAT) is proposed as a tractography algorithm in this study. It is based on self-organizing feature map (SOM), a family in artificial neural networks. The advantage of SOM lies in its ability of mapping high dimensional data into a 1D, 2D, or 3D data space, subject to a topological ordering constraint [
SOM orders the data into meaningful topologies corresponding to the given input data. SOFMAT uses this ability in terms of retaining the underlying structure of the input space and enabling a mapped match of the investigated imaging space resulting in nerve fiber tracts as an output. The final tractography is the converged state of an artificial neuronal map obtained by the iterative synaptic weight update process [
SOFMAT, in an attempt to discover nerve fiber tracts, utilizes an artificial neural network learning scheme inspired by the self-organization in a neurobiological system [
The feature mapping model which SOFMAT implements in this study is arranged in a number of 1D lattice, as described by Kohonen [
SOFMAT inherits unsupervised competitive learning from SOM with the following principles [ The output neurons of the network compete among themselves to be fired, for a given input pattern. Only one output neuron is activated at any one time, called the winning neuron. The winning output node is processed by the self-organization progressing towards the input pattern As an outcome of this self-organized competition and cooperation, the topological connectivity in The input pattern, The input space pattern
In SOFMAT,
SOFMAT identifies the winning neuron by computing a distance function comparing an input pattern
Illustration of the training process. (a) Initial random state of the lattice. The input data vector is displayed here as
Cooperation in SOM algorithm is also inherited in SOFMAT. The level of cooperation of the neighborhood neurons is decided by the winning neuron ( The topological neighborhood
The width,
For a given input pattern
The pixel with the weight
initially there are initial position and orientation of each node for each node of the input pattern, once the winning neuron and its string are determined, a weight update matrix the weight update matrix is computed for that string for updating the orientation information according to the new position as in Figure this procedure is repeated from step (iii) until the maximum number of predefined iteration or convergence is reached; the converging weight matrix that includes the position and orientation information of the multiple strings is the resulting topology of SOFMAT.
The aim of the implementation is to map the underlying topology of a discrete input space. Initially, the weights are assigned randomly and the SOM pattern is arbitrarily positioned. A starting input node is randomly picked among the inputs for training (Figure
For each position update of a node the directional convergence of orientation vector is also achieved.
Following the three processes of the unsupervised learning method, taking into account both the position and the direction of a candidate node, SOFMAT enables the determination of neural fiber tracts having similar diffusivity. The updated neighborhood helps to compute the proper neighbor of each winning neuron, which enables the algorithm to calculate the neural paths with respect to the underlying diffusivity.
The number of inputs is determined by the image dimensions, which is 150 × 150 pixels in PISTE. The number of strings and the number of nodes in each string have been changed between 2–80 and 50–3200, respectively, for experimenting the convergence behavior of different PISTE topologies. The learning rate in (
The tracking results for SOFMAT are shown on four exemplary synthetic data sets. The PISTE trajectories described in Section
In this study, the linear, linear break, orthogonal crossing, and spiral PISTE data sets each of them with individual FA were examined with SOFMAT. Varying FA values give information about the anisotropy and as a result about the anatomy of the tissue investigated. A change in the FA map shows clues about the investigated trajectory. In PISTE, images are created on homogeneous anisotropic background, and decreasing anisotropy along tracts is applied. Therefore, the FA maps serve as filters where the routinely applied homogeneous anisotropic background can be extracted from the image. This process also acts as a noise removal highlighting the diffusion pattern. The eigensystem of
Linear PISTE trajectory. (a) T2 weighted image; (b) input corresponding to the computed eigenvectors (blue). Initial weights
Linear break trajectory. Eigenvectors representing the diffusivity are superimposed on T2 image in blue. SOFMAT results with single string trial are seen in green (right). The gap in the middle of the tract is zoomed to give an idea about the implementation result of the algorithm.
Linear break trajectory with multiple strings. SOFMAT results are seen in blue. The trajectory and the gap in the middle of it are determined by SOFMAT as represented.
Diffusion tensor representation of the orthogonal PISTE trajectory. The different diffusivities for this geometry are seen on the diagonal images. Upper left:
The SOFMAT result superimposed on T2 images for orthogonal crossing trajectory. The tracts are defined along the paths through the total trajectory. Both of the orthogonal tracts are reconstructed completely.
The SOFMAT results are investigated, and the determined neighborhood’s coordinates and their related calculated eigenvectors are evaluated. The histogram shows the similarity of the original input and the SOFMAT’s reconstructed tract for orthocrossing trajectory: noise free (a) and SNR = 5 (b). The angular cost function results inform that the input and the output are nearly the same. Here, the input pattern is 150 × 150, where in both cases noise free and SNR = 5, the parameters
The spiral trajectory results are represented. (a): T2 weighted images; from left to right: noise free T2; SNR = 30; SNR = 15. (b): FA images of each input data. (c): SOFMAT results superimposed on respective T2 weighted images. (d): SOFMAT reconstructions exclusively.
The proposed SOFMAT algorithm is compared with the two well-known fiber tracking suits, GTRACT [
The tracking errors (in mm) of the three tracking tools for linear trajectory.
SLT | GTRACT | SOFMAT | |
---|---|---|---|
SNR = 30 | 0.63 | 0.60 | 0.46 |
SNR = 15 | 0.90 | 0.65 | 0.48 |
SNR = 5 | 1.40 | 0.70 | 0.56 |
The tracking errors (in mm) of the three tracking tools for orthogonal crossing trajectory.
SLT | GTRACT | SOFMAT | |
---|---|---|---|
SNR = 30 | 0.675 | 0.65 | 0.46 |
SNR = 15 | 0.875 | 0.66 | 0.48 |
SNR = 5 | 1.45 | 0.70 | 0.65 |
The mean tracking errors (in mm) of SOFMAT reconstruction of spiral trajectory.
SOFMAT | ||
---|---|---|
Spatial | Angular | |
Noise free | 0.699 | 0.1151 |
SNR = 30 | 0.743 | 0.1127 |
SNR = 15 | 1.611 | 0.3416 |
SNR = 5 | 6.298 | 0.5087 |
The results for all the three tracking tools are represented in Tables
To observe the effectiveness of the ANN based algorithm, the convergence of the cost function is detected as SOFMAT weights are stabilized. The SOFMAT tracking results of an uncertainty region, namely, an orthocrossing trajectory, are also presented in Figure
In all examinations, the input pattern is the T2 weighted image of spiral trajectory with input matrix size of 150 × 150. In order to compare SOFMAT’s results, the network in all of the three cases has 50 strings, where the number of iterations is 6000 and unique in all three exams.
For each of the investigated trajectory, the network’s parameters are methodically and carefully determined. The determination of parameters effect the phases of the network and its ability to converge safe and stably. As mentioned previously, each individual PISTE pattern is examined for a number of iterations. The aim of varying iterations is to find the best match and so to determine the most reliable tract. The more the investigated pattern gets complex, iteration number increases. This also explains why the spiral trajectory’s iteration number (=6000) was the highest among all the trajectories. This is a natural characteristic of a self-organizing network.
In each experiment, the convergence is checked upon both the position and orientational training results. Here, the orthogonal crossing trajectory with 150 × 150 original input size is selected as a sample. In Tables
The validation results of the SOFMAT implementation. The orthogonal crossing trajectory is selected as sample. Here, analysis results for 2-string case are shown.
Node × string |
Mean spatial distance | Angular norm |
---|---|---|
50 × 2 | 5.7429 | 0.1206 |
100 × 2 | 3.5785 | 0.0999 |
150 × 2 | 2.2843 | 0.1003 |
200 × 2 | 1.8195 | 0.0965 |
The validation results of the SOFMAT implementation for orthogonal crossing. A wider network for orthogonal crossing trajectory is investigated and represented.
Node × string |
Mean spatial distance | Angular norm |
---|---|---|
20 × 20 | 8.17 | 0.0923 |
25 × 20 | 3.3412 | 0.0943 |
30 × 20 | 2.8286 | 0.0883 |
40 × 20 | 2.1506 | 0.0910 |
60 × 20 | 1.7862 | 0.0917 |
The validation results of the SOFMAT implementation for orthogonal crossing with updated network parameters.
Node × string |
Mean spatial distance | Angular norm |
---|---|---|
20 × 40 | 1.49455 | 0.0996 |
40 × 40 | 1.1065 | 0.0971 |
60 × 40 | 1.0391 | 0.0969 |
80 × 40 | 0.8346 | 0.0885 |
100 × 40 | 0.6577 | 0.0330 |
150 × 40 | 0.8480 | 0.0890 |
200 × 40 | 0.4263 | 0.0415 |
250 × 40 | 0.3729 | 0.0436 |
300 × 40 | 0.3603 | 0.0504 |
Several studies aim to investigate the synchronization, information transmission, and signal sensitivity in concept of theoretical neuroscience using neural networks [
Unlike the functional magnetic resonance imaging (fMRI), DTI highlights the anatomical connectivity patterns of the brain and does not include any information about the function of the brain activity directly. Generally, clinical DTI studies measure the data voxel wise where the size of each voxel is on the order of millimeters. But it’s well-known that there are millions of fibers passing through each image voxel. Thus, the spatiotemporal dynamics within a voxel can be explored by using the Hudgkin-Huxley neuronal networks [
Our study proposes an artificial neural network approach named SOFMAT based on self-organizing feature mapping to generate simulated tracts based on their diffusivity and to clarify, especially, the fiber tracts in complex fiber architectures. The artificial neuronal topology based on unsupervised learning method is used as a classifier to identify the structured simulated topologies.
Mapping the brain’s white matter noninvasively is possible through proper analysis of DTMR images. The algorithms proposed for fiber mapping and fiber tractography are to be examined by synthetically simulated datasets for accurate validation. In this study, a common synthetic DT dataset, namely, PISTE, which is specially generated for verification purposes of DT and tractography algorithms, is used for verification and validation. One of the main constraints in the accuracy of the mapping results is the determination of intersecting fiber tracts in uncertainty regions. In DTI literature these intersecting regions generate a critical tracking problem. Providing a solution for identification of the orientations of the brain fibers in these uncertainty regions in diffusion tensor analysis is of great importance [
In this study, we proposed a tracking tool for detecting real brain fibers later as a future study according to unsupervised learning method SOFM. The main idea of SOFMAT is to track the complex fibers according to unsupervised learning while keeping the structural information of the underlying tissue. The methodology is applied and examined firstly on computer simulated trajectories PISTE for verification and validation of the algorithm.
The proposed fiber reconstruction method SOFMAT clarifies the diffusivities in the previously mentioned uncertainty regions (Figures
Fiber tracking in SOFMAT begins by identifying seed voxels to be used as potential starting positions for the reconstructed fibers. Based on the predetermined eigensystem of the sample trajectories, fiber tract is estimated within each voxel regarding to the diffusivity defined by this eigensystem. Here, the knowledge in diffusion literature suggests that the eigensystem defines the diffusivity [
The novel algorithm SOFMAT is being evaluated throughout the study. The validation study performed on PISTE gives promising results, and they have been compared to the well-known SLT method and the GTRACT algorithm. In the literature, as it has been represented in [
Investigating samples with both varying noise and different geometry is important for evaluation, because the deviation from the original fiber path is caused mainly by the noise. The relationship between the tracking error to SNR is acceptable in all examinations. Also dependency on the geometry is seen (Figures
The SOFMAT method gives promising results, compared to the traditionally implemented and well-accepted tractography algorithms mentioned above (Tables
This paper represents a novel approach namely self-organizing feature mapping tractography (SOFMAT) for complex fiber tracking purposes in diffusion tensor analysis. The algorithm is based on unsupervised learning in artificial neural networks. As an alternative to the existing methods, SOFMAT is also effective in low anisotropy regions and less affected to noise and curvature complexity than GTRACT and SLT methods. Also, unlike some DTT studies established with PISTE [
The authors have no conflict of interests to disclose.
This work is supported in part by Bogazici University Scientific Research Project no. 07HX104D.