Biomech Model MechanobiolDOI 10.1007/s10237-014-0616-2

ORIGINAL PAPER

Computational simulation of the mechanical response of braintissue under blast loading

Kaveh Laksari · Soroush Assari · Benjamin Seibold ·Keya Sadeghipour · Kurosh Darvish

Received: 12 April 2014 / Accepted: 2 September 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract In the present study, numerical simulations ofnonlinear wave propagation and shock formation in braintissue have been presented and a new mechanism of injuryfor blast-induced neurotrauma (BINT) is proposed. A qua-silinear viscoelastic (QLV) constitutive material model wasused that encompasses the nonlinearity as well as the ratedependence of the tissue relevant to BINT modeling. Aone-dimensional model was implemented using the discon-tinuous Galerkin finite element method and studied withdisplacement- and pressure-input boundary conditions. Themodel was validated against LS-DYNA finite element codeand theoretical results for specific conditions that resulted inshock wave formation. It was shown that a continuous wavecan become a shock wave as it propagates in the QLV braintissue when the initial changes in acceleration are beyond acertain limit. The high spatial gradient of stress and strainat the shock front cause large relative motions at the cel-lular scale at high temporal rates even when the maximumstresses and strains are relatively low. This gradient-inducedlocal deformation may occur away from the boundary andis proposed as a contributing factor to the diffuse nature ofBINT.

Keywords Brain tissue · Quasilinear viscoelasticity ·Hyperelasticity · Blast loading · Discontinuous Galerkinmethod · Injury biomechanics

K. Laksari · S. Assari · K. Sadeghipour · K. Darvish (B)Department of Mechanical Engineering, Temple University,Philadelphia, PA, USAe-mail: kdarvish@temple.edu

B. SeiboldDepartment of Mathematics, Temple University, Philadelphia, PA, USA

1 Introduction

Traumatic brain injury (TBI) is continuously reported as oneof the leading causes of death and disability in the world(Elkin and Morrison 2007). In the USA alone, 1.4 millionpeople sustain a form of TBI each year, of whom 52,000die and over 200,000 are hospitalized. In addition to the TBIcases occurring in motor vehicle accidents and sports relatedinjuries, which are quoted as the major causes of TBI, a rel-atively more recent aspect of TBI, known as blast-inducedneurotrauma (BINT), has also become of increasing concern.BINT is referred to as the signature wound of war in Iraqand Afghanistan wars affecting almost 20 % of the soldiers(Elder and Cristian 2009; Panzer et al. 2011). The mech-anisms of BINT are not yet completely understood and theavailable experimental and computational models to simulateblast injuries are currently limited, signifying the need forimprovement in such models (Cernak and Noble-Haeusslein2010).

A blast wave is a pressure wave with finite amplitude thatin a short time period releases a significant amount of energy.A sharp increase in pressure and air density starts propagat-ing away from the source of explosion. As the blast overpres-sure (BOP) enters a biological tissue, high-rate stress waves,which have longitudinal and shear components, develop thatcan have devastating effects on the tissue depending on thenature of the waves and the type of tissue (Cernak and Noble-Haeusslein 2010). The duration of a blast wave, dependingon the explosion’s intensity, varies in the range of 100µs toa few milliseconds, and it can induce pressure rises of upto 1 MPa (Cernak and Noble-Haeusslein 2010; Moore et al.2009). Such high rates of loading may lead to propagationof acceleration waves and shock waves, due to the nonlin-ear behavior of the tissue, which are not present at low ratesof loading. The nonlinear viscoelastic wave propagation in

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brain tissue has been mostly overlooked in the BINT litera-ture.

Since the pioneering works in the field of nonlinearviscoelasticity in 1960s and 1970s (Coleman et al. 1964,1966; Coleman and Gurtin 1965a, b, c), little attention hasbeen given to the possibility of propagation of accelerationwaves and shock waves in nonlinear viscoelastic soft tis-sues including brain tissue. Based on theoretical results, ina previously undisturbed semi-infinite nonlinear viscoelasticmedium, when certain conditions are met, discontinuities inacceleration, i.e., acceleration waves, and/or in velocity orstrain, i.e., shock waves, can develop and propagate. Thisis counter-intuitive to the expected dissipative character ofviscoelastic media and indeed in the case of linear viscoelas-ticity acceleration waves are always dissipative, i.e., theiramplitudes decay in time and space and do not lead to shockwaves.

While an analytical solution for the propagation of one-dimensional (1D) acceleration waves in nonlinear viscoelas-tic materials exists, the solutions to general displacement orpressure inputs require a numerical solution. For BINT appli-cations, it is necessary to use a numerical method that canhandle the complex constitutive model of brain tissue derivedbased on latest reported results and also the propagation ofacceleration and shock waves in such medium.

The discontinuous Galerkin (DG) method was used in thisstudy as it has been shown in the past to be an efficient methodin studying the propagation of discontinuities in a continuum(Cockburn and Shu 2001; Hesthaven and Warburton 2008).The DG method is an extension of the conventional finite ele-ment (FE) method so it retains the capabilities of FE methodsin handling irregular meshes and a wide range of materialmodels. In addition, due to locality of this method (each ele-ment only communicates with neighboring elements), it hasmuch better potential for parallelization and lower computa-tional cost. The main advantage of the DG method for thisstudy compared to the FE method is its ability to captureshock wave motion and evolution with minimal numericalartifact. In most commercial FE software, a shock wave ishandled by using artificial numerical diffusion in the numer-ical solution and reducing a shock wave to a wave with steepgradient. This is performed by using the bulk viscosity effectexplained in Hallquist (2006). To the best of the authors’knowledge, DG has not been performed as a computationalapproach to study wave propagation in nonlinear viscoelastictissues.

The objective of this study was to put forth a novel DGcomputational method to capture and study the formation andpropagation of shock waves in brain tissue. The effects oftissue characteristics such as nonlinearity and viscoelasticityon the stress waves, in particular steepening of the wave frontin contrast with dispersion and damping, are studied. Thetheoretical rationale together with the numerical results are

then utilized in order to propose an injury mechanism forBINT based on shock wave formation in brain tissues.

2 Governing equations

Since the focus of this study was to understand the effects ofbrain material properties on shock wave formation and prop-agation, a 1D domain for simplicity is considered. Using theconventional continuum mechanics approach, the position ofa material point denoted by X in the undeformed (reference)configuration is described by x = χ(X, t) in the deformedconfiguration. The deformation gradient tensor (F), the rightCauchy–Green strain tensor (C) and the Lagrangian straintensor (E), defined below, were used as measures of defor-mation:

F = ∂x∂X, C = FTF, E = 1

2(C − I) (1)

where I is the identity tensor. For a 1D deformation, Freduces to a single stretch ratio λ = ∂x/∂X , while u = x −Xdenotes displacement, and ε = λ − 1 is used as strain. TheCauchy equation of motion in the Lagrangian descriptionsimplifies to a single equation:

∂P

∂X= ρ0

∂v

∂t(2)

where P is the first Piola–Kirchhoff (engineering) stress, ρ0

is the density of the material in the undeformed configura-tion, and v = ∂u/∂t is the velocity field. For regions wherevelocity and strain are continuous, a compatibility equationcan be written between velocity and strain in the followingform:

∂v

∂X= ∂ε

∂t(3)

Equation (2), together with Eq. (3), can be written as a systemof two first-order partial differential equations (PDE):

∂

∂t

[ε

v

]= ∂

∂X

[v

P/ρ0

](4)

Stress (P) is related to strain (ε) through a constitutive model.In the subsequent sections, a hyperelastic and a nonlinearviscoelastic constitutive model suitable for brain tissue willbe considered and their effects on the development of shockwaves based on the governing equation (4) will be studied,but before that the general conditions for the development ofshock waves is discussed in the next section.

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3 Shock formation criteria

To study shock formation, the governing equation (4) is writ-ten in the following form:

∂

∂t

[ε

v

]+ A

∂

∂X

[ε

v

]= 0 (5)

where

A =[

0 −1−c2 0

]and c(ε) =

√1

ρ0

∂P

∂ε

are the coefficients’ matrix (consisting of the material para-meters) and the stress wave velocity in the medium, respec-tively. ∂P/∂ε is the local tangent modulus that in generalcan be a function of strain ε. In a system of first-order PDEs,as in Eq. (5), characteristic lines are two sets of curves inthe X − t plane aligned with the eigenvectors of the coeffi-cient matrix A, along which the system of PDEs simplifiesto two ordinary differential equations (ODEs) (Logan 2010).The eigenvalues of A give the speed of propagation of wavesalong the characteristic lines. The above system has the fol-lowing eigenvalues and eigenvectors:

Λ = ±c, Q =[−1/c 1/c

1 1

](6)

the values of which depend on c and therefore on ∂P/∂ε. Ina linear material, where the tangent modulus is a constant,the characteristic lines have a constant slope, and therefore,the solution of the ODE along each line remains unique.However, in a nonlinear material where the tangent modulusis a function of strain level, the slope of the characteristiccurves changes accordingly and may result in colliding char-acteristic lines at a particular point in space and time. Whenthe characteristic lines collide, the solution to the PDE is nolonger continuous and is instead defined in the weak sense(Evans 1998). This situation is known as a shock wave dis-continuity, in which the solutions (ε and v) have an abruptjump with large spatial and temporal gradients (infinite intheory). In a stiffening material when the tangent modulushas a positive curvature in tension and a negative curvaturein compression, shock wave discontinuity can evolve from asmooth initial condition. In contrast, in a softening material,a stress wave disperses. The high stress gradients and ratesthat are reached in a stiffening material as the stress wavepropagates in the medium, are proposed to be a mechanismof injury in the case of brain tissue undergoing BOP based onthe constitutive material model presented in the next section.

4 Constitutive model for brain tissue

The constitutive model presented in this section followsthe model reported previously in Laksari et al. (2012). TheLagrangian strain tensor (E) and the second Piola–Kirchhoffstress tensor (S = F−1P) are chosen as the measures ofstrain and stress that satisfy the principle of objectivity. Vis-coelasticity is implemented using a quasi-linear viscoelastic(QLV) model consisting of a convolution integral between astrain dependent instantaneous nonlinear hyperelastic func-tion Se(E), and a time-dependent decaying reduced relax-ation function G(t):

S(t) =∫ t

0G (t − ξ)

∂Se

∂E∂E∂ξ

dξ (7)

For simplicity, brain tissue is assumed to be isotropic and Se

is determined from a compressible generalized Rivlin strainenergy density function W :

W = C10(J1 − 3)+ C01(J2 − 3)

+C11(J1 − 3)(J2 − 3)+ 1

2K (J − 1)2 (8)

The first three terms describe the material behavior in distor-tion (no volume change), and the last term is for dilatation.Parameters C10, C01 and C11 are the Rivlin material para-meters and K is the bulk modulus. J1 and J2 are the first andsecond invariants of the isochoric part of the right Cauchy-Green strain tensor, and J = det(F) is the determinant of thedeformation gradient tensor, which represents change of vol-ume. The 1D instantaneous stress–strain relationship (Pe vs.ε) can be simplified as a polynomial expansion that vanishesat ε = 0:

Pe = pe1ε + 1

2pe

2ε2 + · · · (9)

where

pe1 =

(8

3(C10 + C01)+ K

),

pe2 = −

(56

9C10 + 152

9C01

)

are the first-order (linear) and second-order moduli, respec-tively. It is clear from Eq. (9) that stiffening or softening ofthe material behavior near ε = 0 is dependent on the sign ofp2

e . Also for the material to be physically accurate, the elasticmodulus in this region should be positive (pe

1 > 0) to givepositive energy near ε = 0. As for the rate dependence ofbrain tissue, the following relaxation function in the form ofa Prony series was employed:

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K. Laksari et al.

G(t) =6∑

i=1

Gi e−βi t + G∞ (10)

Characterization of the material properties given in Eqs. (9)and (10), which is suitable for BINT modeling, is describedin detail in Laksari et al. (2014). A summary of the results isgiven here. The elastic response Pe (Eq. 9) was derived as:

Pe(ε) = 2749.41ε − 39271.6ε2 + · · · (kPa) (11)

Equation (11) shows that near ε = 0 the response of braintissue is stiffening in compression but is softening in tension.Therefore, it is expected that shock waves could form forcompressive waves.

The reduced relaxation function G(t) in Eq. (10) wasdetermined as:

G(t) = 0.5894e−1702t + 0.3929e−719t

+ 0.009e−100t + 0.0027e−10t

+ 0.0016e−t + 0.0016e−0.1t + 0.0028 (12)

The first two terms that decay after about 4 ms are the dom-inant terms in Eq. (12) indicating that significant reductionof stress occurs in the first few milliseconds after an abruptloading of brain tissue. How this QLV model would affectthe formation of shock waves in brain tissue is discussed inthe next section.

5 Theory of nonlinear wave propagation in brain tissue

Based on the nonlinear theory of wave propagation in materi-als with memory developed by Coleman et al. (1964) in a 1Dmedium that is initially at rest, an acceleration wave (a wavewith a jump in the acceleration at the wave front but contin-uous in velocity and strain) either disperses or evolves into ashock wave (discontinuity in velocity and strain) dependingon the initial value of the jump and the stress-strain relation-ship of the material. The equation that describes the evolutionof the jump in acceleration α(t) can be written as:

α(t) = αc(αc

α0− 1

)et/τ + 1

(13)

where αc is the critical amplitude, beyond which the acceler-ation wave grows and becomes a shock wave (when α(t) →∞), α0 is the initial value of α(t), and τ the is time con-stant for the change in α(t). The parameters αc and τ aredependent on the material parameters and can be written as:

αc = pe1

pe2

G ′(0)U0, τ = − 2

G ′(0)(14)

where U0 is the wave velocity at zero strain, i.e., U0 =c(ε = 0) = √

pe1/ρ0 and G ′(0) = d G/d t (t = 0) is the

initial slope of the relaxation curve that is negative for realmaterials. For brain material properties presented in Sect. 4the acceleration wave parameters, as reported in Laksari etal. (2014), become U0 = 52.43 m/s, αc = 481.59g (1g =9.81 m/s2), and τ = 1.55 ms. These values are used to verifythe numerical algorithm that is developed in this study fordetermining shock wave formation in brain tissue, which isdescribed in the next section.

6 Discontinuous Galerkin (DG) finite element model

A DG FE algorithm was implemented in this study to inves-tigate the propagation of acceleration waves and formationof shock waves in brain tissue. The formulations used arepresented in this section briefly following the works byCockburn and Shu (1998). The governing equation (4) fora 1D medium can be written in the following vector form:

∂q∂t

+ ∂f (q)∂x

= 0 (15)

where

q =[ε

v

]and f =

[ −v−P/ρ0

]

represent the primary variable and the flux (secondary vari-able) vectors, respectively. After the inner product of theabove equation with N shape functionsψ j (x) and twice inte-grating by parts, the strong DG form of the partial differentialequation can be written as:

∫Dk

(∂q∂t

+ ∂f (q)∂x

)ψi dx =

[(f (q)− f ∗(q)

)ψi

]xkr

xkl

(16)

where Dk represents the kth element of a series of non-overlapping elements that approximate the domain of thenumerical solution. Subscriptions r and l represent the val-ues at the two ends (right and left) of the 1D element. f ∗ is anumerical flux vector that compensates for the difference innodal solutions obtained from adjacent elements. The choiceof the numerical flux function is important for the stability ofthe DG scheme. In this study, the local Lax–Friedrichs fluxwas used that can be written as:

f ∗(q−, q+) = f (q−)+ f (q+)2

+ C

2n̂ · (q− − q+) (17)

where superscripts − and + represent the values in two adja-cent elements and n is the outward pointing normal vector(+1 in the right side and −1 in the left), and C is the mag-nitude of the largest eigenvalue of the matrix A = ∂f /∂q

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Mechanical response of brain tissue

(see Eq. 5), representing the wave velocity in each elementedge. The right-hand side of Eq. (16), for a continuous solu-tion, as expected, will be zero.

Spatial discretization was accomplished using the sameN shape functions for q and f , considering that f (q) cangenerally be a nonlinear function of q:

qk(x, t) =N∑

i=1

q̂ki (t) ψ

ki (x), x ∈ Dk (18)

f k(q(x, t)) =N∑

i=1

f̂ki (t) ψ

ki (x) (19)

where f̂ki (t) = f

(q̂k

i (t))

. In this manner, the combination of

N equations similar to Eq. (16) can be written in the followingsemi-discrete matrix form:

Mki j

dq̂ j

dt+ Sk

i j f̂kj =

[(f k − f ∗)ψi (x)

]xkr

xkl

(20)

where

Mki j =

∫Dkψi ψ j dx, Sk

i j =∫

Dkψi∂ψ j

∂xdx (21)

are the mass and stiffness matrices, respectively. Equa-tion (20) gives a nonlinear system of N equations for Nunknowns q̂ j .

Normalized Legendre polynomials of order n were usedas shape functions (ψn(x) = P̃n−1(x)) with N = 2, i.e.,

qk(r, t) = q̂k1(t) P̃0(r)+ q̂k

2(t) P̃1(r) (22)

where

P̃0(r) = 1√2, P̃1(r) =

√3

2r

and r is a normalized spatial variable with r ∈ [−1, 1] alongeach element. These polynomials provide accurate resultswhen integrated and give a diagonal mass matrix that can beeasily inverted for numerical efficiency (Hesthaven and War-burton 2008; Karniadakis 1999). The integrations in Eq. (21)were performed using the Gauss–Legendre–Lobatto quadra-ture. It is important to note again that the main feature ofthe DG method that is crucial for this application is that thesolution can be discontinuous, i.e., for the same node i, q̂i inthe left element and in the right element can be different. Forthe temporal integration of Eq. (20), a third-order Runge–Kutta explicit scheme was used with adaptive time-steppingbased on the maximum wave velocity in the entire domain. Inaddition, in order to avoid spurious oscillations near the dis-continuities, a generalized slope limiter (ΠN ) was utilized toensure the conservation of physical properties as well as the

1

0 t0

130 mm

1 mm

Time

Dis

plac

emen

t(m

m)

pmax

130 mm

p

0

0 2 4 6Time(ms)

Inci

dent

Pre

ssur

e

Fig. 1 On the left is a schematic of the 1D bar and the applied dis-placement input. The left boundary is compressed for 1 mm at t0 = 2.5and 1.25 ms corresponding to two strain rates (400 and 800 s−1). On theright, a schematic of the 1D bar with the pressure boundary conditionis given, where pmax is the peak overpressure studied for two cases(pmax = 50 and 200 kPa)

formal accuracy of the numerical solution. The ΠN schememakes use of the computational result as a piece-wise Nth-order polynomial and only applies limiting in the elementswhere oscillation is detected, therefore keeps the accuracy inregions of smooth solution (Hesthaven and Warburton 2008).

For numerical simulations, a 1D domain with brain mate-rial properties (a bar) with 130 mm length [average anatom-ical length for human brain in the anterior–posterior direc-tion (Blinkov and Glezer 1968)] was considered as shown inFig. 1. The bar was considered to be fixed at one end (no dis-placement) and free at the other to deform based on the load-ing input. At the free end, displacement and pressure bound-ary conditions were considered. The displacement profile isshown in Fig. 1 on the left, where the maximum displace-ment was 1 mm (0.8 % strain) and the ramp speeds resultedin average 400 and 800 s−1 strain rates. These ramp speedswere chosen based on the predication made from the non-linear theory for the critical amplitude of acceleration waveto cover a range of changes in peak acceleration smaller orlarger than the critical value. In the second set of simula-tions, based on the experimental pressure measurements ina rat BINT model with BOP generated from a shock tubeand pressures measured in the third ventricle (Chavko et al.2007), two pressure profiles were scaled and applied to themodel. The scaling factor of the two pressure profiles wasdetermined to give initial peak pressures (pmax) according tothe lower end (50 kPa) and mid-range (200 kPa) of the fieldmeasurements pertaining to blast induced brain injury as aresult of blast explosions (Zhu et al. 2013).

In the following sections, the results of the numerical sim-ulations obtained for two different material models are dis-cussed. First, only the hyperelastic part of the brain materialmodel was considered. In this case, the input wave is expectedto travel through the material with no decay; however, thesteepening effect of the stiffening material is expected to be

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observed. Secondly, the complete QLV model was used toinvestigate the interaction between the nonlinear and damp-ing effects of the material model.

The spatial resolution for reported results was 9µm, andan adaptive time-stepping was used for temporal integration.Comparison of the h-convergence (keeping the order of thepolynomial constant and successively increasing the numberof elements) and p-convergence (keeping the number of ele-ments constant and increasing the order of the polynomial)with polynomials of up to order 5 (P̃5(r)) showed a linearh-convergence and an exponential p-convergence.

7 Simulation results

7.1 Hyperelastic results

For simulations with hyperelastic material properties, asmentioned above, the material parameters were taken fromthe instantaneous hyperelastic parameters given in Sect. 4. Inorder to validate the DG results, a similar model was devel-oped in LS-DYNA (LSTC, Livermore, CA, USA), whichis a commercially available explicit nonlinear finite elementcode. In the LSDYNA simulation, a three-dimensional (3D)bar was considered and constrained in 2Ds, allowing thedeformation to take place only along one axis with the samespatial resolution as the DG model. The boundary conditionsfor the two ends of the bar were prescribed in the same fash-ion as described above. For the hyperelastic material model,MAT-077-H was used, which simulates a compressible gen-eralized Rivlin material with an additional of Ogden energyfunction to account for the hydrostatic work (Hallquist2006). The resulting strain energy function was identical toEq. (8).

The LS-DYNA code uses the method of bulk viscosity inorder to treat shock waves. In this method, a viscous termis added to the hydrostatic pressure response that smearsshock discontinuities into high gradient but smooth solu-tions and keeps the solution unperturbed far from the shockfront (VonNeumann and Richtmyer 1950; Noh 1976). Thisstep is taken to ensure that the solution remains unique inthe domain and the continuity assumptions of the finite ele-ment method are satisfied. However, as a result, the gradi-ents of strain and stress at the wave front are reduced. In thisstudy, since comparing strain and velocity gradients wereof primary concern, care was taken so that the effect ofbulk viscosity was reduced to a minimum as explained inthe LS-DYNA user’s manual (Q1 = 1.5, Q2 = 0.06). Inorder to give a measure of steepening of the stress waves,the value P ′

max defined as the maximum stress gradient atthe wave front normalized by the initial maximum stressgradient was calculated (Eq. 23). Here, the gradient for a

compressive wave in the shock front is reported as positivenumber.

P ′max =

(max

(∂P

∂x

))t(

max

(∂P

∂x

))t=0+

(23)

While in theory P ′max becomes infinity when a shock wave is

formed, in numerical solutions, due to spatial discretization,this value stays finite. Since in real brain tissue, shock frontsexpectedly have finite thicknesses, P ′

max gives a quantitativemeasure for comparing the steepness of the numerical shockfront for models with equal spatial resolution.

7.1.1 Displacement input

Figure (2) shows the results of DG and LS-DYNA simula-tions to the displacement input at the low and high strain rates(400 and 800 s−1). As can be seen, there is generally a goodagreement between the two simulations in terms of maxi-mum amplitudes. In the slower rate (top row), the stress wavepropagates through the medium with no significant shapechange. In the case of the higher rate (bottom row), the stresswave steepens in the wave front, representing formation ofa shock wave. The DG simulation predicts a higher P ′

max inboth cases. This difference becomes significant in the highrate case after about 1.3 ms, which means that, as the wavepropagates deeper in the model, the DG simulation predicts amuch steeper shock front compared to LS-DYNA. These sim-ulations showed that the applied low strain rate was indeedsubcritical and the high strain rate was supracritical.

The results of this section showed that the DG simula-tions showed a more rapid growth of P ′

max. This effect wasfurther studied for different spatial resolutions from 260 to5µm. This smaller length scale was chosen in order to com-pare the two methods at the cellular level of brain tissue andexamine whether their difference in P ′

max was dependent onthe mesh resolution. For this purpose, the numerical shockthickness as a function of mesh resolution for the 800 s−1

strain rate was calculated as shown in Fig. 3. The numericalshock thickness was defined as the distance between 95 and5 % of the maximum stress. As shown in the figure, if theeffect of the shock wave on cellular features with 10µm is tobe studied, a mesh resolution of 5µm is needed for the DGmethod. The shock thickness with the LS-DYNA model, forthe same mesh resolution, will be more than 100µm, whichmeans that the cellular features effectively will not see anystress gradient. In other words, mesh refinement to model thegeometry of brain cellular features in the LS-DYNA modelis not sufficient to study the effect of shock waves on thesefeatures.

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Mechanical response of brain tissue

Fig. 2 Stress (left column) andnormalized maximum stressgradient P ′

max (right column) asthe wave propagates through thetissue for two strain rates of400 s−1 (top row) and 800 s−1

(bottom row). The results in theleft column are shown for threetime snapshotst ∈ {2, 2.5, 3}ms from left toright respectively

0 50 100 130

0

20

40

60

80

X (mm)St

ress

P(k

Pa)

DGDYNA

0 1 2 2.5

1

15

30

45

t (ms)

Pm

ax

DGDYNA

0 50 100 130

0

50

100

150

X (mm)

Stre

ssP

(kPa)

DGDYNA

0 1 2 2.51

1e4

2e4

t (ms)

Pm

ax

DGDYNA

microstructurelength scale

100 101 102 103100

101

102

103

Mesh Resolution (μm)

Shoc

kFr

ont

Thi

ckne

ss(μ

m)

DG

DYNA

Fig. 3 Numerical shock front thickness as a function of element size(�X ) for the strain rate of 800 s−1. The length scales relevant to braintissue cellular features are shown

7.1.2 Pressure input

In the pressure input simulations (Fig. 4), also the DG andLS-DYNA simulations showed good agreement in the termsof the amplitude of the stress wave. The shape of the wavewas unchanged in the low pressure (50 kPa) input while forthe higher pressure (200 kPa) input the wave front became

significantly steeper with time. In the case of pressure input,similar to displacement input, the stress gradient at the wavefront was significantly higher in the DG model for the higherpressure input.

7.2 Viscoelastic results

The DG model explained in Sect. 6 was used with the QLVmaterial model presented in Sect. 4 to investigate shock waveformation in the viscoelastic brain tissue. Since LS-DYNAmaterial library does not include a QLV model similar to theone described by Eq. (7), the validation of the DG code in thiscase was performed against the theoretical results of acceler-ation waves in nonlinear viscoelastic materials explained inSect. 5. For these validations, a constant velocity input wasconsidered that resulted in an initial jump in acceleration(acceleration wave) applied at the left boundary.

7.2.1 Jump in acceleration

Acceleration waves with subcritical and supracritical initialconditions (α0 = 100, 1,000g respectively) were consideredentering the free end of a long (1 m bar) one-dimensionalQLV model with brain material properties. The evolutionof the acceleration wave shown in Fig. 5, agrees with the

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K. Laksari et al.

Fig. 4 Stress response in thehyperelastic brain material asexposed to pressure inputs with50 kPa (left) and 200 kPa (right)peak pressures at the boundaryfor three time snapshotst ∈ {2, 2.5, 3}ms from left toright respectively

0 50 100 130

0

50

100

X (mm)St

ress

P(k

Pa)

DGDYNA

0 50 100 1300

100

200

300

X (mm)

Stre

ssP

(kPa)

DGDYNA

theoretical predictions given in Eq. (13). For the subcriti-cal initial condition, the acceleration wave is reduced overtime and for the supracritical initial condition, it is increased,which indicates formation of a shock wave. These resultswere considered as validation of the DG QLV model.

7.2.2 Displacement input

The simulation results for displacement input with the QLVbrain material (Fig. 6) show that while the peak amplitude isdecreasing for both rates as a result of viscoelastic damping,the stress gradient at the wave front initially increases in thecase of the supracritical rate before it eventually decreasesand the wave dissipates. This indicates that while the riskof injury due to peak stress is reduced as the stress wave intraveling inside the viscoelastic brain tissue, the risk of injurydue to stress gradient may be increasing. This occurs whenthe initial loading rate surpasses the critical rate and whilethe wave has already penetrated deep into the tissue.

7.2.3 Pressure input

The two pressure levels that resulted in significantly differ-ent steepening behavior of the wave front in the hyperelasticmodel were compared in the QLV model. In the case of pres-sure input, similar to the case of displacement input, the shapeof the wave was changed in the case of supracritical input andwas unchanged for subcritical input (Fig. 7). It is clear fromthis figure that the stress gradient decreases at later times forboth cases. The strain wave is also shown in the right graph.For a viscoelastic material, it is expected that the stress andstrain waves have different trends due to the stress relaxationphenomena. The formation of shock discontinuity is moreevident in the strain waves. However, even strain waves even-tually become smooth as the wave dissipates. This means thathigh strain and stress gradients are reached in a limited timeinterval and therefore a limited spatial domain. Therefore, asin the case of displacement input, it can be concluded that for

a realistic pressure input to brain, a limited spatial domain inthe tissue, away from the exposed boundary, will be at higherrisk of injury due to high strain or stress gradients.

8 A proposed injury mechanism of nervous tissue underblast loading

Based on the reported research over the past several decadeson the mechanisms of TBI as a result of car accidents andsports injuries, three main mechanisms have been proposednamely, intracranial pressure, tissue stress and strain, andrelative motion between brain and skull (Takhounts et al.2008). In recent years, due to improvements in body armorand protection against blast exposures, the number of soldierswho survive with various severity of BINT has increased(Vandevord et al. 2012). Contrary to the brain tissue defor-mations induced in car accidents, where shear deformationplays an important role (Zhang et al. 2004), the deforma-tions created as a result of BOP are volumetric in natureand primarily result in changes in the hydrostatic pressure.Due to the incompressibility of brain tissue, the volumetricdeformations are generally small in magnitude (Feng et al.2013; Humphrey 2003). Also the peak pressure decreases asa pressure wave travels in the tissue due to viscoelastic damp-ing (see Sect. 7.2.3). Therefore, it is yet puzzling how BOPresults in diffuse brain injuries as in the case of shear-inducedTBI. In this study, as introduced in Laksari et al. (2014), it isargued that the one of the characteristics of stress waves thatmay cause injury in the brain is high spatial gradient of stressand strain that may occur at the wave front. The high spatialgradients also occur at high temporal rates, which contributesto their injurious effect.

While it is not yet clear which elements of neural tissuemicrostructure are more vulnerable to biomechanical stressesand deformations, two mechanisms are conceivable for themechanical failure of these elements. First, local stress andstrain exceed certain threshold in a region encompassing

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Fig. 5 Evolution of theacceleration wave for two casesof α0 = 100g (subcritical) andα0 =1,000g (supracritical)initial jumps in acceleration

0 1 2 3 40

25

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time (ms)Ju

mp

inA

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Fig. 6 Viscoelastic stress waveresults for 400 s−1 (left) and800 s−1 (right) strain ratedisplacement inputs for timesnapshotst ∈ {1, 1.5, 2, 2.5, 3}ms fromleft to right respectively

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multiple cellular structures, damaging several microstruc-tural elements. Second, spatial gradients of stress and straingrow quickly, which cause neighboring microstructural ele-ments or the two ends of a single element undergo large rel-ative motions. The latter mechanism that is proposed here asa mechanism of injury, can occur even at stresses and strainsthat are below the injury threshold in the first mechanism.To make this point clearer, a schematic of the nervous tissueincluding axons, neurons, astrocytes, oligodendrocytes, andsmall blood vessels are given in Fig. 8. The length scale ofthese microstructural elements is in the range of 1–30µm(Bear et al. 2007). In such small spatial resolution, the stressgradients that can be induced from propagating dilatationaland shear waves, where no shock is present, will be small. Toaffect this length scales, the waves need to have frequenciesat several MHz–GHz, which would be highly dissipative.However, the waves can evolves into a shock wave, due tomaterial nonlinearity, and create high stress and strain gra-dients relevant to the length scales observed in the nervoustissue microstructural elements. It is shown in the figure thatthe method of bulk viscosity that is used in the LS-DYNAFE code to handle shock waves, is not able to capture thewave front thicknesses as small as the microstructural ele-ments when the mesh geometry is at the length scale of

these elements. This observation regarding the length scaleof the shock front thickness signifies the application of theDG method in handling shock waves in this study. The effectof strain rate in neuronal injury has been shown in previousstudies (Morrison et al. 2010; Effgen et al. 2012; Ellis et al.1995). Therefore, as mentioned earlier, occurrence of highstrain rates due to shock waves can therefore exacerbate theeffect of the relative motion that is caused by the high stressand strain gradients at the shock front and increase the riskof injury.

9 Discussion

For numerical simulation of nonlinear wave propagation inbrain tissue, a constitutive material model is needed thatencompasses the nonlinearity as well as the rate dependenceof the tissue relevant to BINT modeling. To predict the devel-opment of shock waves in brain tissue, two parameters areparticularly important in governing the behavior of an accel-eration wave: the critical amplitude, beyond which an accel-eration wave grows in time, and the decay rate, which deter-mines how fast this growth occurs. These parameters, inturn, depend on the tissue material properties. Specifically, in

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Fig. 7 The left two graphsshow the stress response in theviscoelastic model as exposed topressure inputs with 50 kPa (top)and 200 kPa (bottom) peakpressures at the boundary. Theright graphs show the strainresults for the same loadingcondition, where a shockdiscontinuity in strain is evidentin the 200 kPa case. In allgraphs, four snapshots oft ∈ {1.5, 2, 2.5, 3}ms are shownfrom left to right respectively 0 50 100 130

0

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Pa)

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direction of propagation

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≈ 50 μm

shock front

Fig. 8 Schematic of nervous tissue microstructural elements interacting with a shock front. When the stress gradient is significant at the lengthscale of these elements, it will cause relative motion between the elements that may result in injury. The image was reproduced from Laksari et al.(2014)

addition to the slope of the instantaneous stress-strain curve(tangent modulus), the second-order modulus and the ini-tial rate of reduction of the relaxation function determine theacceleration wave parameters. For brain material properties

that are suitable for BINT applications, the results describedin an earlier work by our group (Laksari et al. 2014) wereemployed. Briefly, in that study a combination of mediumrate finite compression test results and high rate finite torsion

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results were used to derive the material properties of braintissue. Based on those material properties, it was confirmedthat formation of shock waves in brain tissue is possible whenthe initial jump in compressive acceleration is beyond 482g.The growth time constant of the acceleration wave was deter-mined to be 1.6 ms, which indicates that significant growthcan occur before the shock wave amplitude dissipates withinthe length scale of human brain.

The acceleration wave parameters clearly depend on thematerial properties of brain tissue and since determining thematerial properties directly for BINT applications is not atrivial task, it is important to do a parametric study to estimatethe variations in the acceleration wave parameters. Sincebrain is an almost incompressible material, its most debatablematerial property is bulk modulus. In material characteriza-tion and numerical simulations, it is common to assume avalue of 0.49 for the linear Poisson’s ratio of brain tissue(Darvish and Crandall 2001; Hoberecht 2009). As describedin Laksari et al. (2014), our group varied the Poisson’s ratiobetween 0.485 and 0.495 and it was found that the values forthe critical amplitude (319g–996g) still fall within the rangepertaining to blast loading conditions in the field.

In implementation of the DG finite element formulation,a 1D model of brain tissue was studied. This simplificationwas necessary as the complexity of the problem requires astep by step approach and validation of the modeling workat each step. This model was nevertheless sufficient to arguethe main point of this work which was the formation of shockwaves in brain tissue in time scales and length scales rele-vant to BINT. It is expected that the possibility of transfor-mation of compressive waves into shock waves is also validfor 3D. In 3D, there would be two types of waves propa-gating in brain tissue, namely longitudinal (dilatational) andshear (distortional) waves. The two wave types propagate atdrastically different speeds due to brain incompressibility,and therefore, the time scale of their effects may be different.While a blast over pressure may result in primarily longitu-dinal waves in brain tissue, shear waves can also form uponreflections from boundaries (e.g., skull) and tissue interfaces(e.g., white/gray matter, or brain/vasculature). Each of thesewave types can evolve into shock waves due to material non-linearity and result in gradient-induced injuries. A 3D modelto study shock wave formation would therefore require vali-dated representations of the brain tissue boundaries and inter-faces at high loading rates.

While the model studied was finite with fixed boundarycondition at one end, its length was long enough for shocks toform before the compressive wave reached the fixed bound-ary. At the fixed boundary, the stress wave would reflect.Reflection of waves in brain tissue was beyond the scopeof this study and as explained above, a three-dimensionalmodel would be more suitable to study reflection. At the freeboundary, since experimental data for the brain boundary

conditions as a result of blast over pressures entering the headare lacking, two loading conditions that were considered tobe relevant were chosen. In the displacement-input simu-lations (e.g., from compression of the skull), <1 % nominalstrain at different rates was applied with a smooth profile thatresulted in peak accelerations below and above the criticalacceleration. In the pressure-input simulations (e.g., pressurewave passing through the scalp and skull), pressure profilesreported in Chavko et al. (2007) for rat brain were selectedas the basis and the peak pressure was scaled based on blastinjuries reported in pigs (Zhu et al. 2013). Clearly for futurestudies directly measured boundary conditions that result invarious degrees of BINT are needed.

The DG model was validated in two steps. First, it was val-idated for a hyperelastic material model. Although this is aspecial case of the nonlinear viscoelastic material model withno dissipation or constant reduced relaxation function, it gavethe advantage that the DG model could be validated againstcommercial nonlinear finite element codes (LS-DYNA inthis case). As a result, validation could be performed for themodel with finite length and with arbitrary displacement- orpressure-input boundary conditions. Additionally, compari-son of the DG model with the LS-DYNA model showed thatfor capturing shock thicknesses at the length scale of neuraltissue cellular features, the number of elements in the LS-DYNA model would become prohibitively large. This is dueto the limitation of the method of bulk viscosity in smoothinga shock wave discontinuity (Hallquist 2006). To simulate ashock front thickness for neuronal injury, the mesh resolu-tion of the DG model can be in the same order of the cellslength scales while LS-DYNA, for the same mesh resolu-tion, yielded a shock front thickness that was one order ofmagnitude larger. As a result, in the LS-DYNA simulations,individual neurons would not undergo significant stress andstrain gradients as the shock wave passes through them. Thesecond step of the validation of the DG code was with regardto the implementation of the nonlinear viscoelastic (QLV)material model. To this end, since the commercial codes donot have such material model, the only available theoreti-cal prediction (propagation of one-dimensional accelerationwave in a semi-infinite medium initially at rest) was utilized.

Following the validations mentioned above, the DG codewith QLV material model was used to investigate the nature ofshock wave formation for displacement- and pressure-inputboundary conditions. An important result, which was not pre-dicted by the theory, was that the stress and strain gradients atthe shock front initially increased for supracritical conditionsand then decreased. The increase in the gradients occurredwhile the wave amplitude was still significant. Over time, theviscoelastic damping caused the gradient to decrease and thewave to eventually dissipate. The implication of this findingis that while the risk of injury due to high stress or strain con-tinually decreases as the stress wave propagates in the brain

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Fig. 9 Time histories of Pmax(left columns) and P ′

max (rightcolumn) for QLV and linearviscoelastic models at 400 s−1

(top row) and 800 s−1 (bottomrow) strain rates

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tissue, the risk of injury due to high stress or strain gradientincreases for some time period. Therefore, regions in the tis-sue that are away from the boundary may be at higher riskfor gradient-induced injury. This result can partly explainthe diffuse nature of injuries observed in BINT. It is not yetclear whether strain is a better measure for brain injury orstress. In terms of gradients, the simulations showed that atthe shock front, strain gradients were higher, in a normalizedsense, compared to stress gradients. This difference can beexplained based on viscoelastic stress relaxation. It can beconcluded that an injury model that is based on strain gradi-ent, compared to stress gradient, may put the brain cells at ahigher risk.

Variations in peak stress amplitude Pmax and normalizedstress gradient P ′

max at the shock front for the cases of subcrit-ical and supracritical displacement inputs are demonstratedas a function of time in Fig. 9. In this figure, the QLV sim-ulation results are compared with simulation results with alinear viscoelastic material model for brain tissue (ignoringthe second-order modulus). The figure shows that Pmax firstincreases as the displacement input reaches its peak. Due toviscoelasticity, the peak stress occurs after the peak strainand gradually dissipates. The QLV model, as expected (witha stiffening nonlinear stress-strain curve), predicts higher

Pmax than the linear model. The change in P ′max at the shock

front in the QLV model is higher than the linear model andthis difference, particularly for supracritical input, is drastic.The figure shows that a linear approximation would signif-icantly underestimate gradient-induced injuries in brain tis-sue. In the QLV model subject to a supracritical input, P ′

maxincreases significantly within 1.5 ms, which indicates forma-tion of a shock wave. Later, P ′

max decreases as the damp-ing effect of viscoelasticity takes over and dissipates thewave.

Experimental validation of the results presented in thisstudy is currently difficult if not impossible since visualiza-tion of shock waves in a soft solid material is technicallychallenging. An alternative approach would to induce blastbrain injuries in an animal model in which the severity ofinjury can be varied in a controlled manner and compare theobserved injury patterns with computational results. Therewould be challenges also in this approach. For example, notevery detail in the animal brain can be modeled in the compu-tational model. Additionally, the injuries would result fromother mechanisms in addition to high gradient (e.g., highpressure and blood brain barrier breakage). Also the experi-mental pressure measurements will be limited to larger bloodvessels and brain ventricles.

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10 Conclusion

Simulations with the DG finite element code with nonlinearviscoelastic material model developed in this study showedthat, for both displacement- and pressure-input boundaryconditions that represent blast loading conditions, compres-sive stress waves can develop into shock waves in braintissue. Whether a shock wave will form depends on theintensity of the input and brain tissue material properties.A shock wave results in large gradients in stress and strainacross brain tissue cellular features. These large gradientsoccur a few milliseconds after the peak stress and there-fore may occur deep in the tissue away from the incidentsurface. Local cellular high-rate relative motions that aregenerated due to the passage of a shock wave may resultin a type of injury that is termed high gradient-inducedinjury. Gradient-induced injuries would be distinct frominjuries caused solely by high stress or strain amplitudes interms of their location and mechanical deformation of thecells.

Acknowledgments The authors acknowledge funding for this workfrom American Heart Association pre-doctoral fellowship under GrantNumber 11PRE7690042 (K.L.), NHLBI under Grant Number K25HL086512-05 (K.D.), and NSF under Grant Number DMS-1318709(B.S.).

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