By Jillian Kunze
Electrophysiology explores the origins and effects of the electrical properties of biological tissues. This field of inquiry falls into two main groups of methods. Electroencephalography measures the current or voltage that arises due to spontaneous electrical activity in the brain and aims to find the source location, while bioimpedance analysis measures the current or voltage that is caused by outside electrical excitations and aims to find the tissues’ electrical properties. Electrophysiology applications often lead to very ill-posed inverse problems, requiring both hardware and software to be robust to measurement errors and mismodeling.
During a minisymposium presentation at the 2022 SIAM Conference on Imaging Science, Fernando Moura recounted the creation of a statistical anatomical atlas that records the human head’s electrical properties to serve as prior information for cerebral electrophysiology and bioimpedance applications [2]. “To have meaningful and stable solutions, we will need to use prior information,” Moura said. “That was the main objective of this project.” Moura, a researcher from Universidade Federal do ABC in Brazil who is currently spending time at the University of Helsinki in Finland, presented joint work with Leonardo Alves and Roberto Beraldo (both from Universidade Federal do ABC) as well as Samuli Siltanen (University of Helsinki).
Figure 1. A brief description of the process of creating the anatomical atlas’ static component. Figure courtesy of Fernando Moura.
The information in the atlas is four dimensional, with three spatial dimensions in addition to time. Researchers will be able to use the atlas in several ways, including as a statistical prior for their inverse problem solvers, a linearization point in linearization-based algorithms, or a benchmark to test inverse problem solvers. “You can also use the atlas to create statistically representative data,” Moura said. “This is heavily used in machine learning and sensitivity analysis.”
The atlas is composed of a static component that represents the main tissues in the upper part of the human head, as well as a dynamic component for time-varying arterial blood circulation. Both of these parts are assumed to be Gaussian and independent. For the static component, Moura and his collaborators compiled three-dimensional (3D) images created by magnetic resonance imaging (MRI) of 107 individuals that were recorded by CASILab at the University of North Carolina at Chapel Hill.
Figure 1 briefly describes the process of creating the atlas’ static component. The first step is to normalize all of the images to a reference image of an average head MRI, such that they align to the same position and scale. The researchers used the symmetric image normalization method to diminish the differences between each image that may arise due to misalignment or their varying aspect ratios and sizes.
Figure 2. The average results from segmenting all of the images into gray matter (GM), white matter (WM), cerebrospinal fluid (CSF), bones (BO), and other soft tissues (OT). The information in the atlas is three-dimensional, but this figure provides two-dimensional images from the coronal, transversal, and sagittal viewpoints. Figure courtesy of Fernando Moura.
Next, Moura’s group segmented the MRI images for five tissues: gray matter, white matter, cerebrospinal fluid, bones, and other soft tissues. They used statistical parametric mapping—a very common method for segmenting head tissue—and produced an average of the segmentations for different tissues from all patients (see Figure 2). “The images are quite sharp, so it means that segmentations are consistent and very well-aligned,” Moura said. “This is a very good sign that the segmentation really worked.”
Moura and his collaborators assigned the electrical properties of the tissue to each voxel of the segmented images using a frequency-dependent complex relative permittivity model [1]. They then applied methods to perform statistics computations and find the static component of the atlas at different frequencies (see Figure 3).
Moura then moved on to the dynamic component of the anatomical atlas. This aspect needed to contend with the cerebral arteries, which bring blood to different areas of the brain. “Blood, we know, has different electrical properties than other tissues,” Moura explained. “I would like to add this information as well, because we are trying to monitor how the circulation is happening inside the brain — this might be an important prior, and might interfere with the potential that you're measuring if this quantity is changing over time.”
Figure 3. Statistical results for the static component of the atlas. The figure depicts the average at different frequencies for the conductivity (left), resistivity (middle), and relative permittivity (right). Figure courtesy of Fernando Moura.
The researchers segmented the blood vessels through the same general procedure as before with a few changes, using magnetic resonance angiography images from 109 healthy human individuals. “The electrical properties of the blood are a function of the velocity of the blood inside the vessel,” Moura said. “This is a known fact that we see experimentally.”
They solved the Navier-Stokes equations in the superior aortic system to find the cross-sectional pulsatile velocity in each artery, then used that information as input to Visser’s model to find the electrical properties of the blood at each artery along the cardiac cycle. As blood vessel walls are elastic, the model had to account for their influence on the shape of the pressure wave that travels through the vessels and the resulting change in the downstream velocity profiles. The researchers found the flow rate, average velocity, pressure, and resistivity change in different arteries over time. “For each time instant, I can take these values and assign them to each point of the arteries in each one of the patients,” Moura said. “Then I can add this and compute the statistics as I did before, but now for each time instant I have a different average and a new covariance matrix, so that I have this dynamic component of the atlas computed as well.”
Figure 4. The anatomical atlas projected onto a finite element mesh for electrical impedance tomography. Figure courtesy of Fernando Moura.
Now that the atlas is completed with both a dynamic component and static component, researchers can adapt the information that it provides to any desired mesh. Moura showed an example of the atlas projected onto a finite element mesh for electrical impedance tomography (see Figure 4), which researchers could use as a prior to solve problems however they wish.
The code for the Open Source Statistical Anatomical Atlas of the Human head for Electrophysiology Applications as well as precomputed atlases are available online.
References
[1] Gabriel, S., Lau, R.W., & Gabriel, C. (1996). The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues. Phys. Med. Biol., 41(11), 2271
[2] Moura, F.S., Beraldo, R.G., Ferreira, L.A., & Siltanen, S. (2021). Anatomical atlas of the upper part of the human head for electroencephalography and bioimpedance applications. Physiol. Meas., 42(10), 105015.
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Jillian Kunze is the associate editor of SIAM News. |