# Embedded Finite Elements for Modeling Axonal Injury

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## Abstract

The purpose of this paper is to propose and develop a large strain embedded finite element formulation that can be used to explicitly model axonal fiber bundle tractography from diffusion tensor imaging of the brain. Once incorporated, the fibers offer the capability to monitor tract-level strains that give insight into the biomechanics of brain injury. We show that one commercial software has a volume and mass redundancy issue when including embedded axonal fiber and that a newly developed algorithm is able to correct this discrepancy. We provide a validation analysis for stress and energy to demonstrate the method.

## Keywords

Embedded element Volume redundancy Mass redundancy Force redundancy Finite element Brain tissue anisotropy## Notes

### Acknowledgment

The authors gratefully acknowledge the support provided by CFDRC, Inc. under a sub-contract funded by the Department of Defense, Department of Health Program through Contract W81XWH-14-C-0045, the U.S. Army Research Laboratory (ARL) at the Aberdeen Proving Grounds under Contract W15P7T-10-D-D416, the Defense Health Agency under the Contract W81XWH-14-C-0045 and the US Department of Defense through the Contracts W15P7T-10-D-D416, W81XWH-17-C-0216, DOTC-17-01-INIT0086. Neuroimaging data show in Fig. 2 was provided by The Pennsylvania State University Center for Sports Concussion Research and Service, University Park, USA. The authors thank Dr. Sam Slobounov and Dr. Brian D. Johnson for the data provided. We would also like to acknowledge The Pennsylvania State University Social, Life, and Engineering Sciences Imaging Center (SLEIC), High Field MRI Facility for providing access to the imaging equipment. The authors also thank The Pennsylvania State University Institute for Cyberscience for providing the computational resources required for this work.

## References

- 1.Assaf, Y., and O. Pasternak. Diffusion tensor imaging (DTI)-based white matter mapping in brain research: a review.
*J. Mol. Neurosci.*34(1):51–61, 2008.CrossRefGoogle Scholar - 2.Bayly, P. et al. Measurement of brain biomechanics in vivo by magnetic resonance imaging. In: Application of Imaging Techniques to Mechanics of Materials and Structures: Proceedings of the 2010 Annual Conference on Experimental and Applied Mechanics, vol. 4, T. Proulx, Ed. Springer, 2013, pp. 117–128.Google Scholar
- 3.Belytschko, T., W. K. Liu, B. Moran, and K. Elkhodary. Nonlinear Finite Elements for Continua and Structures. New York: Wiley, 2013.Google Scholar
- 4.De Geeter, N., L. Dupré, and G. Crevecoeur. Modeling transcranial magnetic stimulation from the induced electric fields to the membrane potentials along tractography-based white matter fiber tracts.
*J. Neural Eng.*13(2):026028, 2016.CrossRefGoogle Scholar - 5.Murray, D. W., and A. A. Elwi. Nonlinear analysis of axisymmetric reinforced concrete structures. Structural engineering report SER 87 | SER-ID SER87, 1980. https://doi.org/10.7939/R3T14TR1F.
- 6.Fish, J. The s-version of the finite element method.
*Comput. Struct.*43(3):539–547, 1992.CrossRefGoogle Scholar - 7.Fish, J., and T. Belytschko. Elements with embedded localization zones for large deformation problems. In: Computational Structural Mechanics & Fluid Dynamics, edited by A. K. Noor, and D. L. Dwoyer. Pergamon: Elsevier, 1988, pp. 247–256.CrossRefGoogle Scholar
- 8.Garimella, H. T., Yuan, H. Johnson, B. D., Slobounov, S. L., and R. H. Kraft. A two-fiber anisotropic constitutive model of human brain with intravoxel heterogeneity of fiber orientation using diffusion spectrum imaging (DSI). In
*ASME 2014 International Mechanical Engineering Congress and Exposition*, 2014, pp. V003T03A011–V003T03A011.Google Scholar - 9.Garimella, H. T., and R. H. Kraft. Modeling the mechanics of axonal fiber tracts using the embedded finite element method.
*Int. J. Numer. Methods Biomed. Eng.*33(5):e2823, 2017.CrossRefGoogle Scholar - 10.Garimella, H. T., and R. H. Kraft. A new computational approach for modeling diffusion tractography in the brain.
*Neural Regen. Res.*12(1):23, 2017.CrossRefGoogle Scholar - 11.Garimella, V. An embedded element based human head model to investigate axonal injury. PhD Thesis, Pennsylvania State University, 2017.Google Scholar
- 12.Garimella, H. T., A. Przekwas, and R. H. Kraft. Do blast-induced skull flexures result in axonal deformation?
*PloS ONE*13(3):e0190881, 2018.CrossRefGoogle Scholar - 13.Giordano, C., S. Zappalà, and S. Kleiven. Anisotropic finite element models for brain injury prediction: the sensitivity of axonal strain to white matter tract inter-subject variability.
*Biomech. Model. Mechanobiol.*16(4):1269–1293, 2017.CrossRefGoogle Scholar - 14.Gleichgerrcht, E., J. Fridriksson, C. Rorden, and L. Bonilha. Connectome-based lesion-symptom mapping (CLSM): a novel approach to map neurological function.
*NeuroImage Clin.*16:461–467, 2017.CrossRefGoogle Scholar - 15.Guertler, C. A., R. J. Okamoto, J. L. Schmidt, A. A. Badachhape, C. L. Johnson, and P. V. Bayly. Mechanical properties of porcine brain tissue in vivo and ex vivo estimated by MR elastography.
*J. Biomech.*69:10–18, 2018.CrossRefGoogle Scholar - 16.Guy, J., E. A. Ellis, K. Kelley, and G. M. Hope. Spectra of G ratio, myelin sheath thickness, and axon and fiber diameter in the guinea pig optic nerve.
*J. Comp. Neurol.*287(4):446–454, 1989.CrossRefGoogle Scholar - 17.Hagmann, P.,
*et al*. Mapping the structural core of human cerebral cortex.*PLoS Biol.*6(7):e159, 2008.CrossRefGoogle Scholar - 18.Helmut, H. Development of a Continuum-Mechanics-Based Tool for 3D Finite Element Analysis of Reinforced Concrete Structures and Application to Problems of Soil-Structure Interaction. Graz: Graz University of Technology, 2002.Google Scholar
- 19.Iarve, E. V., D. H. Mollenhauer, E. G. Zhou, T. Breitzman, and T. J. Whitney. Independent mesh method-based prediction of local and volume average fields in textile composites.
*Compos. Part Appl. Sci. Manuf.*40(12):1880–1890, 2009.CrossRefGoogle Scholar - 20.Jiang, W.-G., S. R. Hallett, M. R. Wisnom,
*et al*. Development of domain superposition technique for the modelling of woven fabric composites. In: Mechanical Response of Composites, edited by P. P. Camanho. Dordrecht: Springer, 2008, pp. 281–291.CrossRefGoogle Scholar - 21.Johnson, C. L.,
*et al*. Local mechanical properties of white matter structures in the human brain.*NeuroImage*79:145–152, 2013.CrossRefGoogle Scholar - 22.Makarov, S., A. P. Leone, and A. Nummenmaa. Researching fiber networks: computational modeling of complex fibrous tissue geometries.
*IEEE Pulse*8(4):58–61, 2017.CrossRefGoogle Scholar - 23.Mori, S., and J. Zhang. Principles of diffusion tensor imaging and its applications to basic neuroscience research.
*Neuron*51(5):527–539, 2006.CrossRefGoogle Scholar - 24.Ohyama, D., Kurashiki, T., Watanabe, Y., Fujita, Y., and M. Zako. Estimation of mechanical behavior of braided composites based on mesh superposition method. In:
*18th International Conference on Composite Materials, Jeju island, Korea*, 2011.Google Scholar - 25.Shattuck, D. W., and R. M. Leahy. BrainSuite: an automated cortical surface identification tool.
*Med. Image Anal.*6(2):129–142, 2002.CrossRefGoogle Scholar - 26.Sporns, O., G. Tononi, and R. Kötter. The human connectome: a structural description of the human brain.
*PLoS Comput. Biol.*1(4):e42, 2005.CrossRefGoogle Scholar - 27.Tabatabaei, S., and S. V. Lomov. Eliminating the volume redundancy of embedded elements and yarn interpenetrations in meso-finite element modelling of textile composites.
*Comput. Struct.*152:142–154, 2015.CrossRefGoogle Scholar - 28.Tabatabaei, S., S. V. Lomov, and I. Verpoest. Assessment of embedded element technique in meso-FE modelling of fibre reinforced composites.
*Compos. Struct.*107:436–446, 2014.CrossRefGoogle Scholar - 29.Toosy, A. T., O. Ciccarelli, G. J. Parker, C. A. Wheeler-Kingshott, D. H. Miller, and A. J. Thompson. Characterizing function–structure relationships in the human visual system with functional MRI and diffusion tensor imaging.
*Neuroimage*21(4):1452–1463, 2004.CrossRefGoogle Scholar - 30.Werring, D., C. Clark, G. Parker, D. Miller, A. Thompson, and G. Barker. A direct demonstration of both structure and function in the visual system: combining diffusion tensor imaging with functional magnetic resonance imaging.
*Neuroimage*9(3):352–361, 1999.CrossRefGoogle Scholar - 31.Zako, M., Kurashiki, T., Kubo, F., and M. Imura. A multi-scale analysis for structural design of fibrous composites–M3 method. In: 15th international conference on composite materials (ICCM-15), CD ed., Durban, 2005.Google Scholar
- 32.Zhao, W., Y. Cai, Z. Li, and S. Ji. Injury prediction and vulnerability assessment using strain and susceptibility measures of the deep white matter.
*Biomech. Model. Mechanobiol.*16(5):1709–1727, 2017.CrossRefGoogle Scholar