Reuben H. Kraft

Combining the finite element method with structural connectome-based analysis for modeling neurotrauma: connectome neurotrauma mechanics.

This article presents the integration of brain injury biomechanics and graph theoretical analysis of neuronal connections, or connectomics, to form a neurocomputational model that captures spatiotemporal characteristics of trauma. We relate localized mechanical brain damage predicted from biofidelic finite element simulations of the human head subjected to impact with degradation in the structural connectome for a single individual. The finite element model incorporates various length scales into the full head simulations by including anisotropic constitutive laws informed by diffusion tensor imaging. Coupling between the finite element analysis and network-based tools is established through experimentally-based cellular injury thresholds for white matter regions. Once edges are degraded, graph theoretical measures are computed on the “damaged” network. For a frontal impact, the simulations predict that the temporal and occipital regions undergo the most axonal strain and strain rate at short times (less than 24 hrs), which leads to cellular death initiation, which results in damage that shows dependence on angle of impact and underlying microstructure of brain tissue. The monotonic cellular death relationships predict a spatiotemporal change of structural damage. Interestingly, at 96 hrs post-impact, computations predict no network nodes were completely disconnected from the network, despite significant damage to network edges. At early times () network measures of global and local efficiency were degraded little; however, as time increased to 96 hrs the network properties were significantly reduced. In the future, this computational framework could help inform functional networks from physics-based structural brain biomechanics to obtain not only a biomechanics-based understanding of injury, but also neurophysiological insight.

Modeling the mechanics of axonal fiber tracts using the embedded finite element method.

A subject-specific human head finite element model with embedded axonal fiber tractography obtained from diffusion tensor imaging was developed. The axonal fiber tractography finite element model was coupled with the volumetric elements in the head model using the embedded element method. This technique enables the calculation of axonal strains and real-time tracking of the mechanical response of the axonal fiber tracts. The coupled model was then verified using pressure and relative displacement-based (between skull and brain) experimental studies and was employed to analyze a head impact, demonstrating the applicability of this method in studying axonal injury. Following this, a comparison study of different injury criteria was performed. This model was used to determine the influence of impact direction on the extent of the axonal injury. The results suggested that the lateral impact loading is more dangerous compared to loading in the sagittal plane, a finding in agreement with previous studies. Through this analysis, we demonstrated the viability of the embedded element method as an alternative numerical approach for studying axonal injury in patient-specific human head models.

Optimal Pulse Shapes for SHPB Tests on Soft Materials

For split Hopkinson pressure bar (SHPB) tests on soft materials, the goals of homogeneous deformation and uniform uniaxial stress in the specimen present experimental challenges, particularly at higher strain rates. It has been known for some time that attainment of these conditions is facilitated by reducing the thickness of the specimen or by appropriately shaping the loading pulse. Typically, both methods must be employed. Pulse shapes are often tailored to deliver a smooth and sufficiently slow rise to a constant axial strain rate, as this promotes equality of the mean axial stress on the two faces of the specimen, a condition referred to as dynamic equilibrium. However, a constant axial strain rate does not eliminate radial acceleration, which may result in large radial and hoop stresses and large radial variations in the radial, hoop and axial stresses. An approximate analysis (assuming homogeneous deformation and incompressibility) indicates that these radial inertia effects would be eliminated if the radial strain rate were constant. Motivated by this result, we consider loading pulses that deliver a constant radial strain rate after an initial ramp-up. The corresponding axial strain rate is no longer constant on any time interval, but for sufficiently thin specimens the resulting departure from dynamic equilibrium may be small enough to be tolerable. This is explored here by comparing the analytical predictions for the conventional and “optimal” loading pulse shapes with corresponding numerical simulations of SHPB tests on a soft, nearly incompressible material.

A computational analysis of bone formation in the cranial vault in the mouse

Bones of the cranial vault are formed by the differentiation of mesenchymal cells into osteoblasts on a surface that surrounds the brain, eventually forming mineralized bone. Signaling pathways causative for cell differentiation include the actions of extracellular proteins driven by information from genes. We assume that the interaction of cells and extracellular molecules, which are associated with cell differentiation, can be modeled using Turing’s reaction–diffusion model, a mathematical model for pattern formation controlled by two interacting molecules (activator and inhibitor). In this study, we hypothesize that regions of high concentration of an activator develop into primary centers of ossification, the earliest sites of cranial vault bone. In addition to the Turing model, we use another diffusion equation to model a morphogen (potentially the same as the morphogen associated with formation of ossification centers) associated with bone growth. These mathematical models were solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data showing skull bone formation in mouse at different embryonic days in mice carrying disease causing mutations and their unaffected littermates. The results show that the relative locations of the five ossification centers that form in our model occur at the same position as those identified in experimental data. As bone grows from these ossification centers, sutures form between the bones.

A COMPUTATIONAL ANALYSIS OF BONE FORMATION IN THE CRANIAL VAULT USING A COUPLED REACTION–DIFFUSION-STRAIN MODEL

Bones of the murine cranial vault are formed by differentiation of mesenchymal cells into osteoblasts, a process that is primarily understood to be controlled by a cascade of reactions between extracellular molecules and cells. We assume that the process can be modeled using Turings reaction-diffusion equations, a mathematical model describing the pattern formation controlled by two interacting molecules (activator and inhibitor). In addition to the processes modeled by reaction-diffusion equations, we hypothesize that mechanical stimuli of the cells due to growth of the underlying brain contribute significantly to the process of cell differentiation in cranial vault development. Structural analysis of the surface of the brain was conducted to explore the effects of the mechanical strain on bone formation. We propose a mechanobiological model for the formation of cranial vault bones by coupling the reaction-diffusion model with structural mechanics. The mathematical formulation was solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data that provide precise three dimensional (3D) measures of murine cranial geometry and cranial vault bone formation for specific embryonic time points. The results of this study suggest that mechanical strain contributes information to specific aspects of bone formation. Our mechanobiological model predicts some key features of cranial vault bone formation that were verified by experimental observations including the relative location of ossification centers of individual vault bones, the pattern of cranial vault bone growth over time, and the position of cranial vault sutures.

Application of neural networks for filtering non-impact transients recorded from biomechanical sensors

A number of biomechanical sensor applications are available on the market today for sports, as well as military applications. These sensors are capable of measuring accelerations and velocities of impact that may be useful for understanding head and neck biomechanics; although the proper use and accuracy of these analysis techniques is still a topic of research. A challenge with these sensors is filtering the raw data to include authentic impacts and exclude false events that are sometimes registered due to the sensitivity of the sensors. In this study we have developed an algorithm based on artificial neural networks and discrete fourier transform based filtering that will be used to distinguish between a biomechanically relevant event and a non-impact transient which is of no concern. Linear accelerations and angular velocities are used as inputs to the pattern recognition algorithm and the output vector contains the probability of each impact belonging to a certain class. Using this approach we report a specificity of 47% and a sensitivity of 88% in the ability to distinguish between real impacts and non-impact transients.

Validation of Embedded Element Method in the Prediction of White Matter Disruption in Concussions

A better understanding of the axonal injury would help us develop improved diagnostic tools, protective measures, and rehabilitation treatments. Computational modeling coupled with advanced neuroimaging techniques might be a promising tool for this purpose. However, before the models can be used for real life applications, they need to be validated and cross-verified with real life scenarios to establish the credibility of the model. In this work, progress has been made in validating a human head finite element model with embedded axonal fiber tractography (using embedded element method) using pre- and post-diffusion tensor imaging data (DTI) of a concussed athlete. Fractional anisotropy (FA) was used to determine the microstructural changes during injury. These damaged locations correlated well with the damaged locations observed from the finite element model. This work could be characterized as a first step towards the development of a more comprehensively validated human head finite element model.Copyright

Assessing functional connectivity across 3D tissue engineered axonal tracts using calcium fluorescence imaging

OBJECTIVE Micro-tissue engineered neural networks (micro-TENNs) are anatomically-inspired constructs designed to structurally and functionally emulate white matter pathways in the brain. These 3D neural networks feature long axonal tracts spanning discrete neuronal populations contained within a tubular hydrogel, and are being developed to reconstruct damaged axonal pathways in the brain as well as to serve as physiologically-relevant in vitro experimental platforms. The goal of the current study was to characterize the functional properties of these neuronal and axonal networks. APPROACH Bidirectional micro-TENNs were transduced to express genetically-encoded calcium indicators, and spontaneous fluorescence activity was recorded using real-time microscopy at 20 Hz from specific regions-of-interest in the neuronal populations. Network activity patterns and functional connectivity across the axonal tracts were then assessed using various techniques from statistics and information theory including Pearson cross-correlation, phase synchronization matrices, power spectral analysis, directed transfer function, and transfer entropy. MAIN RESULTS Pearson cross-correlation, phase synchronization matrices, and power spectral analysis revealed high values of correlation and synchronicity between the spatially segregated neuronal clusters connected by axonal tracts. Specifically, phase synchronization revealed high synchronicity of  >0.8 between micro-TENN regions of interest. Normalized directed transfer function and transfer entropy matrices suggested robust information flow between the neuronal populations. Time varying power spectrum analysis revealed the strength of information propagation at various frequencies. Signal power strength was visible at elevated peak levels for dominant delta (1-4 Hz) and theta (4-8 Hz) frequency bands and progressively weakened at higher frequencies. These signal power strength results closely matched normalized directed transfer function analysis where near synchronous information flow was detected between frequencies of 2-5 Hz. SIGNIFICANCE To our knowledge, this is the first report using directed transfer function and transfer entropy methods based on fluorescent calcium activity to estimate functional connectivity of distinct neuronal populations via long-projecting, 3D axonal tracts in vitro. These functional data will further improve the design and optimization of implantable neural networks that could ultimately be deployed to reconstruct the nervous system to treat neurological disease and injury.

A new computational approach for modeling diffusion tractography in the brain.

Computational models provide additional tools for studying the brain, however, many techniques are currently disconnected from each other. There is a need for new computational approaches that span the range of physics operating in the brain. In this review paper, we offer some new perspectives on how the embedded element method can fill this gap and has the potential to connect a myriad of modeling genre. The embedded element method is a mesh superposition technique used within finite element analysis. This method allows for the incorporation of axonal fiber tracts to be explicitly represented. Here, we explore the use of the approach beyond its original goal of predicting axonal strain in brain injury. We explore the potential application of the embedded element method in areas of electrophysiology, neurodegeneration, neuropharmacology and mechanobiology. We conclude that this method has the potential to provide us with an integrated computational framework that can assist in developing improved diagnostic tools and regeneration technologies.

Damage Prediction for a Cervical Spine Intervertebral Disc

A large part of the military population develop severe neck pain as a result of complex cyclic loading on the cervical spine. It is hypothesized that this pain is linked to accelerated intervertebral disc degeneration caused by wearing heavier head supported equipments for extended periods of time. This heavy head supported mass exerts high amplitude cyclic loads at the neck that may result in fatigue failure of the intervertebral disc. In this paper, we present a methodology to predict damage in the intervertebral disc over extended periods of time. With this model, we attempt to understand initiation and progression of damage in the disc due to loads exerted on the neck. Such an understanding can be beneficial in the development of better helmets and head mounted equipment for the soldiers.Copyright