Biomech Model Mechanobiol DOI 10.1007/s10237-011-0307-1

ORIGINAL PAPER

An axonal strain injury criterion for traumatic brain injury Rika M. Wright · K. T. Ramesh

Received: 30 November 2010 / Accepted: 22 March 2011 © Springer-Verlag 2011

Abstract Computational models are often used as tools to study traumatic brain injury. The fidelity of such models depends on the incorporation of an appropriate level of structural detail, the accurate representation of the material behavior, and the use of an appropriate measure of injury. In this study, an axonal strain injury criterion is used to estimate the probability of diffuse axonal injury (DAI), which accounts for a large percentage of deaths due to brain trauma and is characterized by damage to neural axons in the deep white matter regions of the brain. We present an analytical and computational model that treats the white matter as an anisotropic, hyperelastic material. Diffusion tensor imaging is used to incorporate the structural orientation of the neural axons into the model. It is shown that the degree of injury that is predicted in a computational model of DAI is highly dependent on the incorporation of the axonal orientation information and the inclusion of anisotropy into the constitutive model for white matter. Keywords Diffuse axonal injury · Traumatic brain injury · Injury criteria · Finite element model · Diffusion tensor imaging · Anisotropic model · TBI · DAI

Electronic supplementary material The online version of this article (doi:10.1007/s10237-011-0307-1) contains supplementary material, which is available to authorized users. R. M. Wright · K. T. Ramesh (B) Department of Mechanical Engineering, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2682, USA e-mail: [email protected] R. M. Wright e-mail: [email protected]

1 Introduction A focused effort is now underway to develop methods for the prevention and treatment of traumatic brain injury (TBI). This is driven in part by the increased number of soldiers sustaining TBI from military incidents and the recent increased public awareness of sports-related TBI. Traumatic brain injury is caused by mechanical loading to the head, such as from impacts, sudden accelerations, and blast loading. The pathology of TBI can range from focal damage of brain tissue to widespread axonal injury. In both mild and severe cases of TBI, one of the most common types of pathology is damage to neural axons, which is classified as diffuse axonal injury (DAI) (Smith and Meaney 2000). Although it is widely accepted that sudden inertial loads to the head cause diffuse axonal injury, the translation of this loading at the macroscale to the damage at the cellular level is still poorly understood. In an effort to better understand this coupling, this work presents a computational model that incorporates the anatomical details of the neural tracts and implements an injury tolerance criterion based on the cellular mechanisms of neural damage. In early TBI studies, injury thresholds for diffuse axonal injury primarily consisted of tolerance criteria that could be measured easily in macroscopic experiments, such as the rotational and translational acceleration of the head and the time duration of inertial loading (Lissner et al. 1960; Versace 1971). An example of such an injury tolerance criterion is the head injury criterion (HIC). Although HIC is commonly used in the automotive industry to predict brain injury, it is often criticized because it is solely based on translational acceleration and does not take rotational acceleration into consideration (Feist et al. 2009). Both translational and rotational accelerations have been shown to contribute to brain injury (Kleiven 2007). As computational models of brain

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trauma were developed, it became possible to compute the stress and strain distributions within brain tissue, and tissue-level injury tolerance criteria were proposed to predict the location and degree of injury. A representative sampling of these tissue-level injury tolerance criteria is presented in Table 1. These measures of injury include pressure, von Mises stress, shear stress, shear strain, and maximum principal strain (Raul et al. 2008). Other metrics of injury based on these measures of stress and strain have also been developed. For example, a cumulative strain damage measure (CSDM), defined as the cumulative volume of brain tissue above a critical strain threshold, has been used as an injury measure for DAI (Bandak and Eppinger 1994; Takhounts et al. 2003). To determine the tissue-level tolerance thresholds for TBI in humans, a relationship must be determined between the loading condition on the head, the TBI pathology, and the tensorial stress and strain states within the brain tissue. Because of the limitations of conducting studies on human subjects, data from a combination of studies must be compared and correlated. These include physical head model studies, finite element models, animal studies, clinical studies, cadaver studies, and accident reconstruction data (Deck and Willinger 2008; Feist et al. 2009; Kleiven 2007; Margulies and Thibault 1992; Meaney et al. 1995; Ommaya et al. 1967; Ueno et al. 1995; Viano et al. 2005; Yao et al. 2008; Zhang et al. 2004; Zhou et al. 1994; Zhou and Schmiedeler 2008). However, variations in the head geometry, material parameters, boundary conditions, loading conditions, and the type

of species often exist between these studies, and all of these variations significantly influence the stress and strain states computed in an analysis (Hrapko et al. 2008). This, in turn, affects the threshold magnitudes chosen for injury tolerance criteria, which is evident by the range of tolerance levels listed in Table 1. In addition to the challenges of handling the variations between studies, the development of injury criteria for diffuse axonal injury is difficult because the damage is not readily visualized with conventional medical imaging modalities. In cases of severe TBI, axonal injury is often accompanied by macroscopic focal damage, such as hemorrhaging, contusions, and tissue lesions, which can easily be detected with medical imaging. However, in cases of mild TBI, where DAI is prevalent, the damage occurs at the cellular and subcellular level, which is generally not visible with in-vivo neuroimaging techniques (Niogi and Mukherjee 2010). Conventional top-down approaches to developing injury criteria for DAI become difficult because of the inability to reliably detect the structural signature of DAI in humans. These top-down approaches relate the loading conditions on the head (at the macroscale) to the axonal pathology and to the stress/strain states within the brain tissue. As an alternative, we suggest a bottomup approach where the injury criterion is built up based on the cellular mechanisms of injury. The injury criterion can then be related to the stress and strain states within the brain tissue and finally to the loading conditions on the head.

Table 1 Examples of tissue level stress and strain injury criteria from the literature Study

Injury type

Injury criterion

Stated tolerance level Method

Yao et al. (2008)

Severe and irreversible TBI

von Mises stress

14.8 ± 4.5 kPa

shear stress

7.9 ± 1.6 kPa

Deck et al. (2008)

50% Probability of mild DAI

von Mises stress

26 kPa

von Mises strain

0.25

Finite element reconstruction of vehicle and pedestrian accidents Finite element reconstruction of vehicle and pedestrian accidents

First principal strain 0.31 50% Probability of severe DAI von Mises stress von Mises strain

33 kPa 0.35

First principal strain 0.40 Kleiven (2007)

50% Probability of mild DAI

Zhang et al. (2004)

50% Probability of mild TBI

First principal strain (corpus callosum) First principal strain (gray matter) Shear stress

Kang et al. (1997)

Severe TBI

von Mises stress

11–16.5 kPa

Strain

0.05–0.10

Margulies et al. (1992) Moderate to severe DAI

123

0.21

Finite element reconstruction of football collisions

0.26

7.8 kPa

Finite element reconstruction of football collisions Finite element reconstruction of motorcycle accident Experimental study on baboon, physical model, analytical simulation

An axonal strain injury criterion for traumatic brain injury

Under inertial loading conditions, brain tissue primarily deforms in shear, and it has been hypothesized that these shear deformations result in the stretching of neural axons (Smith and Meaney 2000). When neural axons are stretched beyond a critical threshold, normal biochemical processes in the cells are disrupted, leading to functional impairment of the neurons or, in severe cases, cell death (Smith et al. 1999). Experimental studies conducted on single axons (Galbraith et al. 1993), nerve fibers (Bain and Meaney 2000), neural cell cultures (LaPlaca et al. 2005), and organotypic brain slice cultures (Elkin and Morrison III 2007) provide support for this cellular mechanism of neural injury. These studies have shown that the degree of electrophysiological impairment and morphological damage of neural cells is directly related to the magnitude and rate of axonal stretch. Implementing an injury criterion that is based on this cellular mechanism of neural damage will improve the predictive capabilities of computational models of traumatic brain injury. Strain-based measures of injury currently available in the literature do not take the anatomical orientation of neural axons into consideration. In this study, we develop a multi-scale modeling approach that accounts for the anatomic structure of brain tissue, and we measure injury using a physiologically relevant injury criterion: axonal strain. This work presents a novel method for measuring diffuse axonal injury in a computational model of TBI. We now have the ability to incorporate local fiber orientations through the use of diffusion tensor imaging, enabling us to construct models on an individual basis. The effectiveness of using an axonal strain injury tolerance criterion for DAI is investigated through a 2-D finite element analysis. The development of this finite element model is outlined in Sect. 2. This includes a description of the anisotropic, hyperelastic strain energy function used to model the white matter and the application of diffusion tensor imaging (DTI) to incorporate the axonal fiber orientations into the computational model. In Sect. 3, the results of the finite element analysis are presented and discussed. It is shown that the inclusion of anisotropy into a model for brain tissue can significantly affect the stress and strain states that develop in the tissue under mechanical loading, which impacts the degree of damage that is predicted in a computational analysis.

2 Methods The continuum framework for this study is delineated in Fig. 1. A multi-scale modeling approach is used to couple the anatomical structure and mechanical behavior of brain tissue at different length scales. At the smallest length scale, a single axon can be defined. The axon is embedded in a matrix consisting of glial cells and interstitial space. The axon level is not explicitly modeled in this study given the inability of current in vivo imaging techniques to resolve down to the scale of a single axon; however, the strain of neural axons at this cellular level is the basis for the injury criterion used in this study. The bundle level of the model consists of a “bundle” of neural axons aligned in a single direction. The direction of neural alignment is dependent on the location within the brain tissue. A transversely isotropic constitutive model is applied at this bundle level to model the behavior of the tissue. Scaling up to the tissue level, it is evident that the orientation of the neural axons varies spatially. The structural orientation of the neural axons is determined through the use of diffusion tensor imaging. A finite element model is constructed at this tissue level and incorporates the information from the bundle and axon levels. This tissue level model is used to analyze the injury response of white matter regions within the brain tissue. Finally, to demonstrate the application of the tissue level model in studying axonal injury under realistic loading conditions at the organ level, a finite element analysis is conducted on a 2-D coronal slice of the head. The head is subjected to inertial loading conditions, and the tissue level FE model is utilized to assess axonal injury in the full head FE model.

2.1 An anisotropic constitutive model for white matter Diffuse axonal injury is commonly located within the deep white matter regions of the brain. Structures such as the splenium of the corpus callosum and the brainstem are cited as being highly susceptible to damage (Smith and Meaney 2000). Unlike the gray matter regions of the brain, which primarily contain neural cell bodies and can be considered isotropic, the white matter consists of an organized arrangement of neural axons and is anisotropic in nature

Fig. 1 Overview of the model length scales. The model bridges the scales from the organ level down to the neural axons at the cellular level

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(Prange and Margulies 2002). Although some finite element studies of brain injury account for the difference in stiffness between the white and gray matter, the directional dependence of the white matter is often assumed to be negligible, and an isotropic model is used for both (El Sayed et al. 2008; Taylor and Ford 2009; Zhang et al. 2004; Zhou et al. 1994). Experimental studies, however, have shown that the material stiffness of the white matter is directionally dependent, especially at large strains (Arbogast and Margulies 1998; Hrapko et al. 2008; Ning et al. 2006; Prange and Margulies 2002; van Dommelen et al. 2010). In a study by Prange and Margulies (2002), a 30% difference in stiffness was found for regions of cerebral white matter with different neural tract alignments at strains up to 50%, and Hrapko et al. (2008) showed that the shear response of the corona radiata is 1.3 times stiffer when measured in the coronal plane compared with the sagittal plane. In this study, the white matter is modeled as an anisotropic material. Nonlinearity is included in the material description through the use of a hyperelastic strain energy function. The use of the hyperelastic model implies that no significant mechanical damage occurs in the tissue, which is an appropriate assumption for strains less than 50%. Prange and Margulies (2002) have shown that no structural changes occur in brain tissue for shear strains up to 50%, and Franceschini et al. (2006) have demonstrated that damage evolution occurs in brain tissue only for strains greater than 50% when loaded under uniaxial tension. The effect of viscoelasticity will not be considered in this analysis to simplify the problem, although it has been shown that the inclusion of nonlinear viscoelasticity can affect the stress response up to 20% and the strain response up to 50% (Brands et al. 2002). 2.1.1 Transversely isotropic hyperelastic model For small representative volumes of white matter and for regions with well-aligned axons, such as the corpus callosum, the white matter can be idealized as a transversely isotropic material. The strain energy function for a transversely isotropic hyperelastic material consisting of one family of fibers is written in terms of the following invariants (Holzapfel 2000):   1 I1 = tr C, I2 = (tr C)2 − tr C2 , I3 = det C, 2 (1) I4 = A · CA, I5 = A · C2 A where C is the right Cauchy–Green deformation tensor and A is a unit vector denoting the fiber direction in the reference configuration (Fig. 2). The invariants I1 , I2 , and I3 represent the isotropic matrix response, and the pseudo-invariants I4 and I5 account for the anisotropic response of the material. To avoid numerical complications associated with the finite element implementation of a nearly incompressible material model, we define a modified deformation gradient F by

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Fig. 2 The deformation gradient, F, relates the fiber direction in the reference configuration, A, to the fiber direction in the deformed configuration, a

F = J −1/3 F

(2)

where J = det F, and we define modified right and left Cauchy–Green deformation tensors : T

T

C = F F, B = F F (3)   The invariants I1 , I2 , . . . , I5 are also defined in terms of these modified deformation tensors (Holzapfel 2000). Assuming that the strain energy is a function of these modified invariants, the following constitutive equation can be derived (Spencer 1972) for a transversely isotropic material 

∂W ∂W ∂W ∂W 2 I+ + I1 B− B σ = 2J −1 I3 ∂ I3 ∂ I1 ∂ I2 ∂ I2   ∂W ∂W  a ⊗ Ba + aB ⊗ a + I4 a ⊗ a + I4 (4) ∂ I4 ∂ I5   where W I1 , I2 , . . . , I5 is the strain energy function, σ is the Cauchy stress, B is the modified left Cauchy–Green deformation tensor, I is the identity tensor, and a is a unit vector denoting the fiber direction in the deformed configuration. As shown in Fig. 2, a is related to the fiber direction A in the reference configuration through the deformation gradient as follows FA = λa

(5)

where λ is the stretch of the fibers (Spencer 1972). 2.1.2 Choice of strain energy function Numerous hyperelastic strain energy functions have been proposed to model the nonlinear, isotropic behavior of brain tissue (Bilston et al. 2001; Brands et al. 2002; El Sayed et al. 2008; Franceschini et al. 2006; Meaney 2003; Miller and Chinzei 1997; Prange and Margulies 2002). However, few strain energy functions have been developed to model the anisotropic behavior of white matter. In Velardi et al. (2006), a modified version of the Ogden model was used to describe the anisotropic response of white matter, and a quadratic reinforcing strain energy function has previously been used to

An axonal strain injury criterion for traumatic brain injury Table 2 Viscoelastic material properties from finite element human head models in the literature Density (kg/m3 )

Study

Material

Kuijpers et al. (1995)

White and gray matter 1,040

8.3e−3

338,000

169,000

50–10,000

Willinger et al. (1999)

White and gray matter 1140

2.19

528,000

168,000

35

Takhounts et al. (2003)

White and gray matter –

0.56

10,300

5,000

100

Zhang et al. (2004)

White matter

2.19

41,000

7,800

400

Kimpara et al. (2006)

1,040

Bulk modulus Short-term shear Long-term shear Decay constant (GPa) modulus (Pa) modulus (Pa) (s−1 )

Gray matter

1,040

2.19

34,000

6,400

400

Brainstem

1,040

2.19

58,000

7,800

400

White matter

1,000

2.16

12,500

6,100

–

Gray matter

1,000

2.19

10,000

5,000

–

Brainstem

1,000

2.19

23,000

4,500

–

Kleiven (2007)

White and gray matter 1,040

2.1

12,500

1,000

–

Yao et al. (2008)

Cerebral tissue

1,060

2.19

12,500

2,500

80

Cerebellum

1,060

2.19

10,000

2,000

80

Brainstem

1,060

2.19

22,500

4,500

80

White matter

1,040

2.19

16,400

6,800

–

Gray matter

1,040

2.19

13,600

8,200

–

White and gray matter 1,040

El Sayed et al. (2008) Watanabe et al. (2009)

Taylor and Ford (2009) White matter Gray matter

2.19

12,500

2,500

80

1,040

2.37

41,000

7,800

700

1,040

2.37

34,000

6,400

700

model the brainstem (Ning et al. 2006). Anisotropic formulations of the Fung and Ogden model were compared with a structurally based anisotropic model for white matter in an analysis by Meaney (2003). In this study, the quadratic reinforcing model is applied to model cerebral white matter. The reinforcing strain energy function has the following form: W =

 K0 2 θ G0  I1 − 3 + I4 − 1 (I3 − 1)2 + 2 2 2

(6)

where G 0 is the shear modulus, K 0 is the bulk modulus, and θ is a measure of the additional reinforcing provided by the fibers. This simple form of the reinforcing model allows for easy computational implementation while containing enough information to capture the anisotropic constitutive behavior of white matter. The first two terms of the strain energy function (the extended Neo-Hookean model) account for the isotropic behavior of the white matter, where G 0 and K 0 are an effective shear and bulk modulus of the axonal fibers and matrix material. The anisotropic behavior is captured through the last term of the strain energy function, where θ is a measure of the additional stiffness provided by the alignment of the neural axons in the white matter. 2.2 Material properties The bulk modulus and density values for white matter were chosen to be consistent with those values commonly used

in finite element models of the head (Table 2). An estimate for the remaining material parameters for the white matter in this study is obtained by fitting the transversely isotropic hyperelastic model of Eqs. 4, 5, and 6 to tensile data from the experimental work of Velardi et al. (2006). Velardi et al. subjected white matter sections of porcine brain tissue to uniaxial tensile loading in directions parallel and perpendicular to the preferred fiber orientation. Our model is fitted to tests performed on the corona radiata by Velardi et al., and the following estimates were determined for the shear modulus and the   fiber reinforcement  parameter:  G 0 = 286 Pa R 2 = 0.905 and θ = 121 Pa R 2 = 0.987 . The quality of the fit to the experimental data is shown in Fig. 3. We note that the shear modulus value determined from the fit to Velardi’s experimental study is a couple of orders of magnitude lower than values commonly used in computational studies of TBI as shown in Table 2. The discrepancy results from the fact that most computational studies base their material parameters on early experimental studies on brain tissue such as Estes and McElhaney (1970), Fallenstein et al. (1969), Galford and McElhaney (1970), and Shuck and Advani (1972). The stiffness of brain tissue cited in these early studies is generally a couple of orders of magnitude larger than that determined from more recent experimental studies (Hrapko et al. 2008; Prange and Margulies 2002; Shen et al. 2006; Velardi et al. 2006). The differences in the material parameters could be due to variations in tissue

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Fig. 3 Plot showing the fit of the material model (solid lines) to the experimental data from Velardi et al. (2006) (round markers)

preservation, experimental protocol, and postmortem time (Hrapko et al. 2008). Since this major difference in shear modulus would make it difficult to compare our results with those in the literature, the long-term shear modulus value from Zhang et al. (2004) was adopted as the shear modulus value in our finite element analysis. Results using the much smaller G 0 from Velardi are presented in Online Resource 1-A; the primary results of this manuscript are not changed by using the Velardi shear modulus. The fiber reinforcement parameter, θ , was then recomputed using the θ/G 0 ratio derived from the Velardi study. Table 3 summarizes the material parameters used here. 2.3 Application of diffusion tensor imaging to derive fiber orientation Diffusion tensor imaging (DTI) offers an in vivo method for determining the orientation of neural axons within the Table 3 Material parameters for white matter used in this study

Fig. 4 Map of the fractional anisotropy (left) and fiber orientation (center) for a coronal slice of the human brain. The finite element analysis was conducted for white matter regions labeled A–D (Head image on right from http://www. secondpicture.com)

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human brain by measuring the diffusion of water molecules within the brain tissue. Since water molecules diffuse faster in the direction parallel to neural fibers than perpendicular to them, the average orientation of neural axons for a given region of the brain can be computed (Le Bihan et al. 2001). The axonal fiber orientation information for this work was obtained from diffusion tensor imaging data from a population-averaged atlas. This atlas was developed from tensor maps of 81 normal subjects (M: 42, F: 39, average age: 38 (18–59 years old), right-handed). The nominal resolution of the imaging matrix was 2.5 mm. Details of this atlas can be found in Mori et al. (2008). From the diffusion tensor imaging data, the fractional anisotropy (FA) and fiber orientation map were obtained for a coronal slice of the human brain (Fig. 4). The fractional anisotropy is a normalized measure of the degree of directionality of the diffusion, and it is computed from the eigenvalues of the diffusion tensor (Le Bihan et al. 2001). It ranges from a value of zero, which indicates that the diffusion is uniform in all directions, to a value of one, which indicates that the diffusion is along a single direction (Niogi and Mukherjee 2010). Since the directionality of the diffusion correlates with the direction of fiber alignment, the fractional anisotropy can also be used to measure the degree of fiber alignment, and it is a useful tool for segmenting the white matter from the gray matter (Niogi and Mukherjee 2010). In the fractional anisotropy map in Fig. 4, the lighter regions represent tissue with a larger degree of fiber alignment, i.e., the white matter, whereas the darker regions represent tissue that is isotropic, i.e., the gray matter. The color-coded fiber orientation map indicates the average fiber orientation of neural axons for each voxel (1 mm3 ) of the image. Red represents neural axons traveling in the right–left direction; blue represents axons in the superior–inferior direction; green represents axons in the anterior–posterior direction. Four small representative regions of white matter were chosen for the analysis. The letters superimposed on the FA and fiber orientation maps

Density, ρ (kg/m3 )

Bulk modulus, K 0 (GPa)

Shear modulus, G 0 (Pa)

Fiber reinforcement, θ (Pa)

1,040

2.0

6,400

2,716

An axonal strain injury criterion for traumatic brain injury

correspond to the regions of white matter that were analyzed in this study. These include regions within the corpus callosum (Region A), the corona radiata (Region B), the boundary between the corpus callosum and corona radiata (Region C), and the posterior limb of the external capsule (Region D). Each region of interest represents a 7 mm2 sample of white matter. These regions were chosen for their varying degrees of fiber anisotropy and for their high susceptibility to diffuse axonal injury (Niogi and Mukherjee 2010).

2.4 Axonal strain injury criterion In recent years, there have been numerous experimental studies aimed at defining injury thresholds for the functional damage of neural cells (Bain and Meaney 2000; Elkin and Morrison III 2007; Geddes and Cargill 2001; Morrison et al. 2003; Wolf et al. 2001). These injury thresholds are measures of stress or strain at either the cellular or tissue level. To define an axonal injury criterion, we use an experimental study by Bain and Meaney (2000). Bain and Meaney stretched the optic nerve of a guinea pig in-situ at strain rates of 30– 60/s and analyzed the resulting functional and morphological injury due to stretch. An “optimal” strain threshold (which optimized both the specificity and sensitivity measures) of 18% was determined for the onset of electrophysiological impairment. This tissue-level threshold of 18% is adapted as the axonal strain injury tolerance criterion in our study and is used as an indicator for the onset of functional damage. Although the injury criterion developed by Bain and Meaney is formally at the tissue level, it is highly representative of the cellular mechanisms of axonal injury. In the optic nerve, the axons are highly aligned in a single direction; therefore, stretching the nerve along its axis induces stretch of the individual axons within the nerve fiber. As the axons are stretched, structural and biochemical changes occur in the cells that eventually lead to axonal disconnection and degeneration. The degree of axonal damage was determined by Bain and Meaney by measuring changes in the electri-

Fig. 5 The fiber orientation map is obtained from diffusion tensor imaging (left). The average fiber orientation is shown in red for each DTI voxel within a region of interest (center). Each voxel contains hundreds

cal potentials generated by the neural cells and by staining for common markers of the axonal pathology. Although the magnitude of cellular strain is not identically equal to the tissue-level strain measured by Bain and Meaney due to the undulated nature of the neural axons, these strain measures are proportional to one another, and the tissue-level strain threshold is a representative measure of the axonal damage that occurs due to axonal stretch. To implement the injury criterion into our model, each white matter region of interest is divided into voxels as shown in Fig. 5. Each voxel is assigned a fiber orientation based on the DTI data. Given the limited resolution of DTI, the fiber orientation is actually averaged over all axons within a 1 mm3 voxel, and the neural axons are assumed to be perfectly aligned in a single direction within a voxel. In our finite element implementation of the model, each voxel is further subdivided into elements, and the axonal strain threshold is applied at this element level. The axonal strain is defined as the component of the nominal strain resolved in the fiber direction for a given element. Stretching of the element in the direction of fiber alignment induces stretching of the neural axons, and the level of stretch is then related to axonal damage through the axonal strain injury criterion.

2.5 Finite element model A two-dimensional plane strain finite element analysis was conducted for each region of interest through the use of the Abaqus 6.9-EF/Standard commercial software package (Dassault Systèmes Simulia Corp., Providence, RI, USA). The appropriate mesh density for the analysis was determined through a mesh convergence study. A total of 784 (4-noded) bilinear elements were used to model each region. The orientations of the neural axons were incorporated into the model by projecting the fiber orientation vectors from the diffusion tensor imaging onto a two-dimensional plane. The fiber vectors were then interpolated to achieve

of neural axons that are assumed to be aligned in a single direction within an element (right). The transversely isotropic constitutive model can then be applied to each element

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3 Results and discussion 3.1 Fiber distributions

Fig. 6 Finite element mesh for a representative white matter region (on right). Fiber orientations are overlaid on the mesh in red, and the applied stretches, λ1 and λ2 , are along x1 and x2 coordinate axes, respectively

the higher spatial resolution corresponding to the element size, and the appropriate fiber orientation was assigned to each element of the finite element mesh based on location as shown in Fig. 6. The quadratic strain energy function was implemented into the model through a user subroutine in the finite element software.

2.6 Applied loading In a full head finite element simulation, the white matter regions of the brain are subjected to complex loading conditions. However, to develop an understanding of the injury response of white matter, it is convenient to first analyze the injury response under a simple loading condition. In this study, each white matter region is subjected to a biaxial stretch state. Displacement boundary conditions are applied to the outer surfaces of the regions of interest (Fig. 6). The stretches λ1 and λ2 correspond to stretches in the x1 and x2 directions, respectively. The stretch is defined as the ratio of the final length over the initial length, and the strain is defined as the ratio of the change in length over the initial length. A linear ramp in displacement was applied for varying ratios of displacement in the x1 and x2 directions, and the axonal strain was computed by resolving the strain in the fiber direction for each element. The 2-D analysis was conducted under a plane strain condition. While a plane strain or plane stress analysis does not accurately represent the full 3-D behavior of the tissue, these approximations can be used to determine bounds on the behavior of the tissue. For the plane strain biaxial stretch problem used in this study, the tissue is highly constrained, and the hydrostatic response dominates the problem. This has a significant effect on the stress states within the tissue, but it does not significantly affect the injury response computed with the axonal strain injury criterion (see Online Resource 1-B).

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As shown in Fig. 6, a finite element mesh was created for each white matter region of interest, and the fiber orientations were extracted from diffusion tensor imaging. Figure 7 shows the fiber orientations for each white matter region of interest. The angular distribution of axonal fibers in each region is shown as a histogram beside each fiber map. These angles were computed from the interpolated fiber angles used in the finite element analysis. For the corpus callosum (Region A), the fibers are primarily oriented along the x1 direction. The average fiber orientation angle is −5.0 degrees as measured counterclockwise from the x1 -axis, and the standard deviation of fiber angles is ±11.3 degrees. The fibers are bimodally distributed, with the majority of the fibers oriented at angles either above or below the primary fiber direction. Note, however, that fiber orientations are constant within each element. The fibers of the corona radiata (Region B) and the external capsule (Region D) are primarily oriented in the x2 direction and have average fiber angles of 99.7 and 73.2 degrees and standard deviations of ±5.7 and ±4.8 degrees, respectively. Thus, within a neural tract (Regions A, B, and D), the axonal fibers are primarily oriented in a single direction; however, at a boundary region between two neural tracts (Region C), there is a greater variation in the angular distribution of the fibers. For the boundary region between the corpus callosum and corona radiata (Region C), the average fiber angle is 74.5 degrees, and the standard deviation of fiber angles is ±17.4 degrees, which is largest of the four regions. In addition, the fibers are more uniformly distributed over a larger range of orientation angles as compared with the other white matter regions. 3.2 Injury maps The global stretch values that correspond to the onset of injury for each white matter region are represented by the curves plotted on the left in Fig. 8. The onset of injury is defined as the condition at which the axonal strain for any single element anywhere within the domain first exceeds the axonal injury threshold value of 18%. These injury threshold curves bound the safe domains for each region. The shape of the curve is highly dependent on the average fiber orientation and the angular distribution of fibers within each white matter region. For example, the fibers in the corpus callosum (Region A) are primarily oriented along the horizontal x1 -axis; therefore, the region is much more susceptible to axonal damage when stretched in the x1 direction. Much larger stretches can be sustained in the x2 direction before the injury threshold is exceeded. This is apparent from the injury threshold curve (Fig. 8a). The onset of injury occurs

An axonal strain injury criterion for traumatic brain injury Fig. 7 On the left, fiber orientations are shown for the four representative regions of white matter (a corpus callosum; b corona radiata; c boundary between the corpus callosum and the corona radiata; d posterior limb of the external capsule). On the right, the corresponding angular distribution of fibers is shown for each region. The fiber angles are measured counterclockwise about the x-axis

at a stretch value of 1.17 when stretch is applied only in the x1 direction; however, a much larger stretch value of 2.16 is achieved before the onset of injury when stretched solely in

the x2 direction. In contrast, the fibers in the corona radiata (Region B) and the external capsule (Region D) are primarily oriented the vertical x2 direction; therefore, these regions

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Fig. 8 On the left, the stretch values (λ1 and λ2 ) that correspond to the onset of injury are plotted for each region of interest (a corpus callosum; b corona radiata; c boundary between the corpus callosum and the

corona radiata; d posterior limb of the external capsule). On the right, a map of the area fraction of injured tissue is shown

are much more susceptible to injury when stretched in the x2 direction as compared with the x1 direction. Although the boundary region between the corpus callosum and the corona radiata (Region C) contains a wider distribution of fiber orientations as compared with the other three regions, the primary fiber direction is in the x2 direction. The onset

of injury in this region occurs at lower stretch values in the x2 direction as compared with the x1 direction. The onset of injury plot is a useful tool in determining the bounds between safe and injurious stretch states; however, it is often useful to assess the degree of injury within a tissue sample. This can be analyzed by computing the area fraction

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of damaged tissue. A map of this area fraction of injury is plotted for each white matter region of interest in Fig. 8. The injury map is color coded to represent the area fraction of tissue that exceeds the axonal strain injury threshold of 18% for each region, where the dark blue color indicates that the entire tissue region is below the injury threshold and the dark red indicates that the entire region exceeds the threshold of injury. In general, as expected, a larger percentage of the white matter becomes damaged as the magnitude of stretch is increased. The map of the area fraction of injured tissue can be used to determine which loading paths are most likely to cause damage to the neural axons, and it can be used to relate the mechanical loading to the probability and degree of injury within a tissue sample. By extending this analysis technique to encompass an entire fiber tract region of the white matter, the degree of damage within the fiber tract can be computed, and these measures can be used to relate cognitive impairment to fiber tract damage. This extension will be presented in a subsequent paper. 3.3 Effect of choice of injury criterion The injury predictions in such computations are highly dependent on the choice of injury criterion. This is apparent by comparing the anisotropic modeling results (left column)

Fig. 9 The von Mises stress (top row), maximum principal strain (middle row), and axonal strain (bottom row) are plotted for Region C for the following stretch state: λ1 = 1.13 and λ2 = 1.23. The results are

for the boundary region (Region C) in Fig. 9. The von Mises stress, maximum principal strain, and axonal strain are plotted for imposed stretch values of λ1 = 1.13 and λ2 = 1.23 as an example. The distribution of injured tissue differs significantly between these measures of injury. For example, an injury criterion based on the maximum principal strain would indicate that injury would initiate on the lower left-hand side of Region C because this is the location with the largest magnitude of strain; however, based on the axonal strain and von Mises stress measures, injury would initiate on the upper right-hand side. In general, the maximum principal strain is a poor measure for diffuse axonal injury because it does not account for the anatomical arrangement of axonal fibers. The axonal strain, on the other hand, takes this axonal orientation information into consideration. Mendis (1992) also proposed the use of an injury criterion based on the structural orientation of neural axons. To predict axonal injury in the corpus callosum, Mendis used an oriented strain measure, defined as the strain component along the direction of the axons in the corpus callosum; however, this injury criterion was just applied for the corpus callosum. The approach proposed in this work can be applied to all regions of white matter. In addition to analyzing the location of injury, the degree of injury for an applied stretch can also be compared for the different injury criteria. This degree of predicted injury

shown for both an anisotropic model for white matter (on left) and an isotropic model for white matter (on right)

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is significantly affected by the magnitude of the injury tolerance threshold. For the results showed in the left column of Fig. 9, 89% of Region C is predicted to be injured with the axonal strain injury criterion using a strain threshold of 0.18. Using the maximum strain injury threshold of 0.21 from Kleiven (2007), which was used to indicate a 50% probability of mild DAI in the corpus callosum, axonal damage would be predicted in 99% of the elements of Region C. On the other hand, the maximum strain injury threshold of 0.31 from Deck and Willinger (2008) would result in a 0% prediction of injury for Region C. Thus, the degree of injury is highly dependent on the chosen injury tolerance threshold. The von Mises stress values computed in Fig. 9 are much lower than the von Mises stress injury thresholds cited in the literature (Deck and Willinger 2008; Yao et al. 2008; Zhang et al. 2004). This is primarily due to the use of a plane strain condition in this analysis, even though the shear modulus can also affect the magnitude of the von Mises stress. If the analysis was conducted using plane stress elements, the magnitude of von Mises stress would be larger and would be comparable with those stresses associated with diffuse axonal injury. A number of injury criteria have been proposed for DAI, and this study demonstrates the importance of selecting an injury criterion that has a physical or physiological basis. In addition, the tolerance threshold for a given injury criterion has a significant effect on the predicted degree of injury. Further work is required to validate the effectiveness of the axonal strain injury criterion proposed here for predicting the location and degree of axonal injury. Studying diffuse axonal injury in humans presents a challenge because the loading conditions that lead to injury are often unknown, and the pathology cannot be readily visualized with in vivo imaging techniques. Therefore, to validate the axonal strain injury criterion, an animal model should be used. Animal models, such as porcine, primate, sheep and rat, have commonly been used to study diffuse axonal injury (Anderson et al. 2003; Cernak 2005; Margulies and Thibault 1992; Raghupathi et al. 2004; Smith and Meaney 2000; Wang et al. 2010). One advantage with animal models is that the loading condition on the head can be highly controlled, and the axonal pathology can be examined through a histological analysis. The anatomical regions of injury can then be compared with injury regions predicted with a finite element model using an axonal strain injury criterion. A correlation between these injury regions would be useful to support the use of the axonal strain injury criterion to predict DAI. 3.4 Effect of anisotropy on response and injury Lanir (1983), Hurschler et al. (1997), Billiar and Sacks (2000), Freed et al. (2005), and Gasser et al. (2006) have developed methods of handling the anisotropic behavior of soft tissue by representing the distribution of fibers through

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an orientation density function. The orientation density function is defined for each family of fibers in the tissue and is incorporated into the constitutive model either through a structure tensor (Gasser et al. 2006), splay invariant (Freed et al. 2005) or through a summation of the strain energies of the individual fibers (Lanir 1983; Billiar and Sacks 2000). In the work of Freed et al. (2005) and Gasser et al. (2006), a hyperelastic transversely isotropic framework is developed to model the dispersion of fibers about preferred fiber orientations in the tissue. A similar approach is adopted in this study to model the anisotropic behavior of white matter. Instead of defining the transversely isotropic model at the tissue level, the constitutive model is defined locally at the element level. A primary fiber orientation is computed for each element based on the DTI fiber orientation map. An orientation density function can also be defined at this element level using the fractional anisotropy data from DTI; however, in this study, it is assumed that the degree of fiber alignment is high for each element and the local dispersion of the fibers is ignored. The global spatial distribution of fibers, however, is incorporated in the model by defining an average orientation for each element of the model. This approach offers a method of coupling the internal microstructure of the tissue to its mechanical response. The inclusion of anisotropy into a material model for white matter has a significant effect on the predicted degree of axonal damage. As an example, the von Mises stress, maximum principal strain, and axonal strain are plotted for Region C for stretch values of λ1 = 1.13 and λ2 = 1.23 (Fig. 9) for both anisotropic (left column) and isotropic (right column) constitutive models for the white matter. In the anisotropic model, the quadratic reinforcing strain energy function was used to model the white matter as described earlier. For the case of the isotropic model, the white matter was modeled as a Neo-Hookean material with a shear and bulk modulus consistent with that used for the anisotropic model. Modeling the white matter as an isotropic material results in a uniform distribution in the von Mises stress and maximum principal strain. In contrast, a spatial variation in the stress and strain is evident when the contribution of the fiber stiffness is included (in the anisotropic material model). These differences in the stress and strain can have important implications for the prediction of injury when a stress or strainbased injury tolerance criterion is used. In the case of the isotropic model, the entire sample region will be labeled as either injured or uninjured depending on the injury threshold level; however, given the anisotropic nature of the white matter, the stress and strain distributions are not uniform and there is a distribution of injury. Some locations will exceed the injury threshold before other locations for a given loading condition, and this location-dependent susceptibility to injury is not captured with the isotropic model. By applying

An axonal strain injury criterion for traumatic brain injury

an anisotropic model for the white matter, the location of injury can be resolved to a greater degree. In contrast to the von Mises stress and maximum principal strain distribution, the distribution of axonal strain can be captured reasonably well even with an isotropic constitutive model if the fiber orientation information is used from the DTI. Similar to the anisotropic model, the axonal strain is computed by resolving the strain in the direction of the axonal fibers as obtained from the diffusion tensor imaging. Note, however, that the contribution of the fiber stiffness is not included in the isotropic model. For stretches of λ1 = 1.13 and λ2 = 1.23, 89% of Region C is predicted to be injured with the anisotropic model whereas 96% is injured with the isotropic model. Reducing the stretch values to λ1 = 1.11 and λ2 = 1.20, injury is predicted in 64 and 72% of the region with the anisotropic and isotropic models, respectively. The area fraction of injury correlates relatively well between these models (percent difference < 12%). Although the additional stiffness due to the fibers is not incorporated into the isotropic model, the axonal fiber orientation must still be tracked to compute the axonal strain. That is, the anatomical detail can be incorporated into a computation through the axonal strain injury criterion alone, rather than also through the constitutive model, and still obtain reasonable results (at least with these material parameters). The strain distribution is resolved reasonably well with the isotropic model; therefore, in this instance, it is not necessary to use an anisotropic constitutive model for white matter to approximate the probability of injury with an axonal strain injury criterion. This simplification may offer the advantage that there are fewer material parameters that need to be determined to implement the material model and axonal injury criterion computationally.

To simulate a realistic loading condition for TBI, the head model was subjected to a sudden rotation about the coronal plane (Fig. 10a). The head was rotated about a point 6 cm below the base of the skull at a constant angular acceleration of 16,000 rad/s2 for 14 ms, corresponding to 90 degrees of rotation. This angular acceleration was chosen because it falls within the injury regime for DAI, according to a criterion proposed by Margulies and Thibault (1992). To analyze the axonal injury that results from this inertial loading, the strain is resolved along the x1 and x2 directions (the coordinate directions rotate with the overall rigid body motion of the head) for a given region of white matter in the full head FE model. Given the strain history, the loading path can be plotted in stretch space, and the degree of injury can be determined from tissue-level injury plots, such as those presented in Fig. 8. In Fig. 10b, the loading path is plotted for the corona radiata region (Region B). Due to this inertial loading, 73% of Region B is predicted to sustain axonal damage. As is evident from the loading path, a larger fraction of the corona radiata will be injured if the duration of applied acceleration is increased. A similar analysis can be conducted on the other three regions of white matter, resulting in injury

3.5 Extension to a full head model The finite element analyses presented in this study were conducted on small regions of white matter under simple loading conditions; however, these results can be applied to assess axonal injury in a full head finite element model. To demonstrate this extension to a full head model, a 2-D finite element plane strain model of a coronal slice of the head is constructed. The model includes the skull, dura mater, falx cerebri, cerebrospinal fluid (CSF), ventricles, white matter, and gray matter. Material properties were taken from the literature (Zhang et al. 2001). The skull, dura mater, and falx cerebri were treated as linear elastic materials. The white and gray matter were modeled as isotropic, viscoelastic materials using large deformation theory. The properties of the CSF were assumed to be similar to water. The hydrostatic response of the CSF was modeled using a Mie-Gruneisen equation of state, and a shear viscosity was defined to model the deviatoric behavior.

Fig. 10 Extension to a full head model (a) a coronal angular acceleration is applied to a 2-D FE head model, and injury is analyzed within a 7 mm2 region of the coronal radiata (Region b). b The loading path (white line) due to this applied rotational inertial loading is plotted in stretch space for the corona radiata region. According to the injury map, 73% of this region will sustain axonal damage due to the applied inertial loading

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predicted in 22% of the corpus callosum region (Region A), in 15% of the boundary region (Region C), and in 0% of the external capsule region (Region D). This submodeling approach can be applied to predict the degree of axonal damage in any white matter region of interest. An advantage of using this approach is that a high level of structural detail does not have to be incorporated into the full head FE model. The structural detail (i.e., the anisotropic orientation of neural axons) is included in the subscale FE model of the white matter region of interest, and the injury plot corresponding to the subscale model is used to determine the degree of axonal injury in this region based on its deformation in the macroscale full head FE model.

4 Summary Axonal strain has been shown to be a mechanism that leads to the functional damage of neurons, but it has not been used as a measure of injury in computational models of traumatic brain injury. The advantage of using such an injury criterion is that it offers a method of coupling the cellular mechanisms of damage with the mechanical loading at the macroscale. This injury criterion was implemented into a finite element model through the use of diffusion tensor imaging, and it was shown that the injury response of white matter is dependent on the primary orientation and the angular distribution of axonal fibers. The difference in the injury maps between regions of white matter demonstrates the importance of incorporating axonal orientation information into a measure of injury. In addition, the inclusion of anisotropy into a constitutive model for white matter has a significant effect on the predicted injury locations when tissue-level measures of injury such as the von Mises stress and maximum principal strain are adapted as injury criteria. For the specific case of an axonal strain injury criterion, the location and degree of injury can be approximated reasonably well with an isotropic constitutive model since the injury criterion takes into account the microstructural arrangement of the neural axons. In the development of the multi-scale modeling framework used in this study, several assumptions were made to simplify the analysis and to specialize it to the problem of interest. The white matter was assumed to be a hyperelastic material. This is a reasonable assumption since our definition of injury is based on functional damage, which occurs at relatively low strains. However, mechanical damage has been shown to accumulate in brain tissue at strains greater than 50%. Therefore, the constitutive model should be extended to include damage parameters to model the behavior at these large strains (El Sayed et al. 2008). Furthermore, viscoelasicity was not included in the subscale FE models. Under loading conditions applicable to traumatic brain injury, brain tissue is subjected to a wide range of strain rates (Saraf et al.

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2007). To accurately model the progression of damage, the constitutive model should be extended to account for the time-dependent behavior of brain tissue. Finally, the injury criterion was based on the stretch of neural axons; however, it has been shown that neural damage is also dependent on the rate of loading (Elkin and Morrison III 2007; Geddes and Cargill 2001; LaPlaca et al. 2005). Therefore, in future studies, the effect of strain rate on the functional damage of neural cells should be incorporated into a measure of injury. This study investigates one of the many possible injury mechanisms for traumatic brain injury: axonal stretch. Other mechanisms of injury include increases in intracranial pressure, local compression, or pinching of neural axons, tissue tears, blood vessel rupture, cavitation, etc. Of all of these injury mechanisms, damage to neural axons is the most prevalent pathology in TBI (Tang-Schomer et al. 2010). Accurately modeling axonal damage in a computational model is extremely important for the model to be a viable tool in developing preventative measures for TBI. This study offers a novel approach for including the anatomical structure of white matter into a computational model so that a physiologically relevant injury criterion, the axonal strain, can be implemented as a measure of injury. Acknowledgments We gratefully acknowledge Dr. Susumu Mori at the Johns Hopkins University School of Medicine for providing access to the diffusion tensor imaging data used in this study. This work is supported by a National Science Foundation Graduate Research Fellowship, the Center for Advanced Metallic and Ceramic Systems, and NIH grants P41RR15241 and R01AG020012.

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