FAConstructor: an interactive tool for geometric modeling of nerve fiber architectures in the brain

The manual input of three-dimensional coordinates allows a precise definition of fiber pathways. A different radius can be assigned to each coordinate, allowing each object to grow or shrink along its trajectory. In addition, the user can apply cubic spline interpolation to smooth the fiber object. In this case, both coordinates and radii are interpolated.

A major advantage of the developed tool is that individual fiber objects can be modified without need to regenerate the whole model. Both whole fiber objects and individual fiber coordinates can be rotated, translated, or duplicated. In this way, the user can modify an already existing fiber model and build large-scale models without much effort. Furthermore, it is possible to group several fiber objects together and change or duplicate them at the same time. The groups can be exported for future usage.

When filling a fiber bundle with fibers, the cross section with minimum radius is used as starting point. The fibers are arranged on a triangular grid (see Fig. 2b), and the radius of every fiber and the minimum distance between neighbored fibers are defined by the user. The triangular grid is rotated from one data point to the next using the tangents of each point and computing the rotation angle using the Rodrigues formula [5].

If the radius increases along the course of the fiber bundle, the distance between the fibers increases accordingly. This guarantees that the fibers in the bundle do not intersect with each other, provided that the user-defined minimum distance does not fall below the diameter of the fibers. The generated fibers are stored as a group, allowing efficient modifications of the fiber bundle.

Coordinates and radii of the fiber objects are defined as single-precision floating-point values, which can be saved either in a hierarchical file system-like data format (HDF5, https://​www.​hdfgroup.​org) or, alternatively, in a plain text file format. HDF5 is advantageous, because it allows to handle metadata, it has no limitation on the number or size of data objects in the collection, giving great flexibility for big data, and it provides high-performance I/O with a rich set of integrated performance features.

As the FAConstructor allows to generate fiber objects interactively, an efficient rendering of the data is crucial. Therefore, a new frame is only rendered when the user interacts with the fiber objects by rotating or translating them, allowing fluent interaction. Rendering is realized with the OpenGL 3.3 framework of Qt5 [4, 25] which enables the use of widgets for applying options and generating fibers, while rendering is done in a single central widget. In addition, OpenMP [22] can be used optionally for several tasks granting performance improvements. Fiber objects and individual fibers are approximated by cylindrical shapes using the defined coordinates and radii. As circular objects cannot be rendered without approximation, they are represented by n-sided regular polygons. Even when displaying a large amount of fibers, the tool maintains a sufficient frame rate: The parameter n is dynamically adjusted to ensure a frame rate above 30. In addition, double-buffered V-Sync is used to reduce stutter which might occur when encountering a fluctuating frame rate. Nonvisible fiber objects are culled out using the Cohen–Sutherland algorithm [8]. As the fiber object trajectory is a line until drawn in the OpenGL shaders, this algorithm can be applied. The user is able to interact with the rendering area by moving, rotating, and scaling the shown content. When losing focus of the current object, the user can reset the view, displaying the whole object in the center. It is also possible to display different orthogonal views of the model and walk through the model using keyboard shortcuts. The user is able to hide certain fiber objects or whole groups by deselecting the checkboxes next to their corresponding labels (cf. Fig. 2b). This helps to keep an overview and improves performance.

Fiber bundles are colored according to their respective orientation in three-dimensional space, using either an RGB (red–green–blue) or an HSV (hue–saturation–value) color map which are also used to visualize 3D-PLI data shown in Fig. 1. In the right bottom corner of the rendering area, a color sphere displays the orientation of the fiber objects. The sphere changes when the user rotates the model. To distinguish individual fibers, it is also possible to assign each fiber bundle a randomized color based on the fiber index.

Figure 3 shows the measured performance of the program. The corresponding datasets were created with the following parameters:where x(t), y(t), z(t) are space coordinates and r(t) is the radius of the fiber object at this position. The functions describe fiber objects consisting of \(N-1\) segments. This allows to conduct performance measurements as a function of the number of objects.

The test procedures to generate Fig. 3a, b, d consisted of starting the program, creating or loading the corresponding data points, and measuring the time or frame rate. Afterward, the program was closed to avoid possible influence on the following iteration. This procedure was repeated ten times for each data point. Measuring the performance when filling the fiber bundles was done by creating a dataset containing n fiber objects with 1000 data points each using Eq. 1. The fiber objects were filled with fibers with a distance of \(d = 0.2\) and a radius of \(r = 0.1\), yielding 21 fibers. The measured time of 20 separate runs was taken and averaged. As the program can use OpenMP for parallelization, both variants were measured individually.

As seen in Fig. 3a–c, the program tends to scale linearly with the produced amount of data points. The usage of OpenMP shows a speedup of 1.21 to 2.1 depending on the measured part of the program. The frame rate in (D) maintains the 30 fps target when the amount of fiber points lies below 1.6 million. This is due to the insufficient performance of the graphics card to display all data points simultaneously. Even when rendering around 60 million data points, 11 GiB RAM and 2.3 GiB video memory are consumed, indicating that neither is a limiting factor. When rendering, the CPU usage stays around 0% to 10%.

3D-PLI is a postmortem imaging technique for unstained histological brain sections. Due to the birefringence (optical anisotropy) of the nerve fibers, the state of polarization changes when light passes through brain tissue. In 3D-PLI, this change of polarization is measured with a polarimetric setup (polarizer, specimen, quarter-wave retarder, analyzer), allowing to determine the spatial orientations of the nerve fibers with micrometer resolution. The measurement and signal analysis of 3D-PLI have been described in detail by Axer et al. [2]. As the investigated brain sections are about \(60\,\upmu \hbox {m}\) thick, each measured volume element contains several nerve fibers. 3D-PLI extracts only the predominant fiber orientation, and the orientations of individual nerve fibers within deep white matter structures can therefore not be determined. Inhomogeneous nerve fiber structures, like crossing fibers or regions with varying fiber densities, influence the measured birefringence signal and may lead to misinterpretations in the reconstructed fiber orientations.

Matrix calculus (simPLI) The in-house developed software simPLI [6] computes the change of polarization of a light beam when passing through a brain tissue sample. The light beam is modeled by Stokes vectors, the optical elements of the polarimeter and the brain tissue by Müller matrices [21]. The birefringence of the nerve fibers is modeled by optical retarder matrices, with the retarder axis oriented along the fiber axis [16]. To account for the imaging system, the image is downsampled and blurred with a Gaussian function with \(\sigma = 0.714\,\Delta {x}\) as described by Menzel et al. [16]. Finally, noise is modeled according to Schmitz et al. [23]. The resulting signals are analyzed using the ROFL algorithm [23], resulting in fiber orientation maps.

Finite-Difference Time-Domain (FDTD) simulation To study more complex light–matter interactions like scattering and interference, Finite-Difference Time-Domain (FDTD) simulations are used. With FDTD simulations, it is possible to study light transmitted through artificial nerve fiber models as well as scattering of light [18, 19]. The propagation of the electromagnetic light wave through the sample was computed by \(\textit{TDME3D}^{{\mathrm{TM}}}\), a massively parallel three-dimensional Maxwell Solver based on a conditionally stable FDTD algorithm [20, 28]: The electromagnetic field components are computed numerically by discretizing space and time and approximating the spatial and temporal derivatives in Maxwell’s curl equations by second-order central differences. More details about the algorithm can be found in Menzel et al. [17].

Matrix calculus (simPLI) The collision-free Fiber Cup phantom was used as input for simPLI (see “Simulation methods” section). The fibers were modeled as homogeneous materials with an absorption of \(\mu = 5\hbox { mm}^{-1}\), and the birefringence was modeled with a strength of \(\Delta {n} = 0.001\) and one axis of anisotropy oriented along the longitudinal fiber axis [16]. For the simulation, a section of \(60\,\upmu \hbox {m}\) thickness from the mid-plane of the Fiber Cup phantom was used, discretized into voxels of \(1\,\upmu \hbox {m}\) side length. The simulations were performed with wavelength \(\lambda = 525\hbox { nm}\), ingoing light intensity \(I_{{0}} = 26{,}000\), and image pixel size \(\Delta {x} = 20\,\upmu \hbox {m}\).

Figure 5a shows the resulting fiber orientations as a vector field, overlaid with the original fiber model. In regions with straight fiber bundles, the fiber orientations are in good agreement with the underlying fiber structure. In crossing regions, however, some fiber orientations are misinterpreted. In region (3), where both crossing fiber bundles contain the same amount of fibers, the resulting vector field shows the average orientation of the two underlying fiber bundles (pink). If fiber tractography, algorithms were applied to the vector field, and the crossing fibers (\(\times \)) could be misinterpreted as kissing fibers (\(> <\)). In region (1), the resulting vector field is dominated by the orientation of the fiber bundle (green) that consists of three bundles (nos. 4, 5, and 6, see Fig. 4). The orientation of the other fiber bundles (purple) is not visible in the resulting vector field so that the crossing of the fiber bundles is not detected. In region (2), the three fiber bundles (4, 5, and 6) join each other. Bundles 4 and 5 (green) have similar orientations and are already indistinguishable when they merge with bundle 6 (red). The resulting orientation vectors of bundle 6 assimilate with the orientation vectors of the other two bundles. The original course of bundle 6 in the crossing regions is not visible in the resulting vector field.

Finite-Difference Time-Domain (FDTD) simulation The three selected regions of the Fiber Cup phantom in Fig. 5a were also studied with FDTD simulations to investigate the scattering of light. For this purpose, a volume of \(128 \times 128 \times 60\,\upmu \hbox {m}^3\) was selected in each region and the propagation of the light wave through the sample was simulated as described in the “Simulation methods” section. The simulations were performed for uniaxial perfectly matched layer absorbing boundaries with a thickness of \(1\,\upmu \hbox {m}\), a Yee mesh size of 25 nm, a duration of 400 periods, a Courant factor of 0.8, and normally incident light with 550 nm wavelength (modeling the polarizing microscope) [18, 19]. All tissue components of the sample (fibers and surrounding tissue) were modeled by dielectric materials with real refractive indices (neglecting absorption). Each fiber was modeled by an inner axon and a surrounding myelin sheath with two layers, as described by Menzel et al. [18]. The simulations were performed on the supercomputer JURECA [11] at Forschungszentrum Jülich GmbH, Germany. Figure 5b shows the resulting scattering patterns, i. e. the intensity per wave vector, for the three selected regions. The scattering patterns reveal the underlying fiber structure and crossing angles of the fiber bundles. The light is always scattered under angles perpendicular to the principal axis of the corresponding fiber bundle. For reference, the perpendicular axes are indicated by straight colored lines around the scattering patterns. (Colors were chosen according to the colors of the fiber bundles shown in Fig. 5a.) In contrast to simPLI, the FDTD simulations allow to resolve individual fiber orientations also in regions with crossing fibers. The determined angles correspond to the angles of the Fiber Cup phantom (cf. Figs.  4 and 5b). Bundles with a larger amount of fibers (regions (1), (2), in green) cause stronger scattering than bundles with a smaller amount of fibers.