A micromechanical hyperelastic modeling of brain white matter under large deformation

J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 2 ( 2 0 0 9 ) 2 4 3 – 2 5 4

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Research paper

A micromechanical hyperelastic modeling of brain whitematter under large deformation

G. Karami∗, N. Grundman, N. Abolfathi, A. Naik, M. Ziejewski

Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, USA

A R T I C L E I N F O

Article history:

Received 18 April 2008

Received in revised form

21 August 2008

Accepted 22 August 2008

Published online 30 August 2008

A B S T R A C T

A finite element based micromechanical model has been developed for analyzing and

characterizing the microstructural as well as homogenized mechanical response of brain

tissue under large deformation. The model takes well-organized soft tissue as a fiber-

reinforced composite with nonlinear and anisotropic behavior assumption for the fiber

as well as the matrix of composite matter. The procedure provides a link between the

macroscopic scale and microscopic scale as brain tissue undergoes deformation. It can be

used to better understand how macroscopic stresses are transferred to the microstructure

or cellular structure of the brain. A repeating unit cell (RUC) is created to stand as a

representative volume element (RVE) of the hyperelastic material with known properties

of the constituents. The model imposes periodicity constraints on the RUC. The RUC is

loaded kinematically by imposing displacements on it to create the appropriate normal and

shear stresses. The homogenized response of the composite, the average stresses carried

within each of the constituents, and the maximum local stresses are all obtained. For each

of the normal and shear loading scenarios, the impact of geometrical variables such as

the axonal fiber volume fraction and undulation of the axons are evaluated. It was found

that axon undulation has significant impact on the stiffness and on how stresses were

distributed between the axon and the matrix. As axon undulation increased, the maximum

stress and stress in the matrix increased while the stress in the axons decreased. The

axon volume fraction was found to have an impact on the tissue stiffness as higher axon

volume fractions lead to higher stresses both in the composite and in the constituents.

The direction of loading clearly has a large impact on how stresses are distributed amongst

the constituents. This micromechanics tool provides the detailed micromechanics stresses

and deformations, as well as the average homogenized behavior of the RUC, which can be

efficiently used in mechanical characterization of brain tissue.c© 2008 Elsevier Ltd. All rights reserved.

d

1. Introduction

Computer simulations are increasingly being used to studytraumatic brain injuries and also for the modeling and

∗ Corresponding author. Tel.: +1 701 231 5859; fax: +1 701 231 8913.E-mail address: [email protected] (G. Karami).

1751-6161/$ - see front matter c© 2008 Elsevier Ltd. All rights reservedoi:10.1016/j.jmbbm.2008.08.003

simulation of neurosurgical procedures. Accuracy of thesesimulations is highly dependent upon the accuracy of thematerial properties used to represent the materials thatcompose the tissue. Traumatic brain injury occurs when a

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244 J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 2 ( 2 0 0 9 ) 2 4 3 – 2 5 4

physical trauma causes damage to the brain and the mostcommon and devastating type of such injuries is diffuseaxonal injury (DAI) (Taber and Hurely, 2007; Saatman et al.,2008). DAI occurs as a result of shearing forces that take placewhen the head is rapidly accelerated or decelerated. Theseshearing forces create microscopic damage to axons in thewhite matter of the brain and cause focal lesions to developin the brainstem and corpus callosum. The stretching andtearing or the swelling of axonal fibers are characteristics ofDAI (Morrison III et al., 2006). Due to the devastating nature ofthis injury, study of axon and extracellular matrix structurehas acquired great interest in biomechanical engineeringresearch. Bain and Meaney (2000) studied the tissue levelthresholds for axonal damage in central nervous systemwhite matter, but it is not yet known how the macroscopicstresses and strains are transferred to the structure of thetissue on the microscopic scale.

Soft biological tissues, including the brain tissue, canbe modeled as anisotropic, nonlinear hyperelastic materials(Usal et al., 2005; Fung, 1981). In the brainstem andwhite matter of the brain, embedded within a matrix ofextracellular components and neurons are bundles of axonalfibers (or neural tracts) which are highly uniaxially orientedand these are believed to provide the strengthening effect(Arbogast and Margulies, 1999). Arbogast and Margulies(1998) performed oscillatory shear tests on adult porcinebrainstems and found them to have a significantly transverseisotropic behavior. The corpus callosum and corona radiataare white matter in the brain and they were tested indifferent directions by Prange and Margulies (2002). Theaxonal fibers within the corpus callosum are highly uniaxiallyoriented as they connect the two hemispheres of the brain.Brain tissue is a complex tissue to model as test resultsfrom a number of sources vary greatly with differences intesting conditions, sample preparation, regional locations,loading directions, and loading rates. In vitro, experimentshave also been conducted on the tissue in uniaxial loading(Miller, 2001; Miller and Chinzei, 1997, 2002; Bilston et al.,1997, 2001; Prange and Margulies, 2002). In vivo, testinghas been performed through the use of indentation testsby Miller et al. (2000) and Gefen and Margulies (2004). Thetesting in the above mentioned experiments finds mainly theoverall response of the tissue but not much microstructuralinformation on the tissue samples. Prange and Margulies(2002) and Velardi et al. (2006) arranged their test samplesso that the axonal fibers were oriented parallel to orperpendicular to the direction of loading, but they did notprovide information about the axonal fiber densities withinthe samples.

The relationship between brain tissue’s stresses andstrains and how they are related to axonal injury havebeen closely studied by several researchers in microscopicscale. Bain and Meaney (2000) developed a strain thresholdfor axonal damage. Analytical microstructural models weredeveloped by Bain et al. (2003) to study how the undulationof axons changed as strain was applied to brain tissue.The microstructural models were used to describe differentlevels of coupling between the axons and the surroundingmatrix. Pfister et al. (2006) developed a testing device forthe uniaxial stretching of axons to study axonal injury and

neural cell death. The device is able to reach strains greaterthan 70% at several loading rates. Arbogast and Margulies(1999) developed a fiber-reinforced composite model of thebrainstem to model its viscoelastic behavior in shear. Meaney(2003) presented a model to represent highly oriented whitematter and compared it with three energy based hyperelasticmaterial models and experimental data. His model was usedfor large deformations and also considered the undulation ofaxonal fibers.

In this study, a finite element based micromechanicalmodel has been developed for analyzing and characterizingthe mechanical response of brain tissue under large defor-mation. Well-organized soft tissue is modeled as a hypere-lastic fiber-reinforced composite with a micromechanics toolas it is used to study the tissue’s nonlinear, anisotropic be-havior. Each of the constituents in the composite tissue isrepresented by an isotropic hyperelastic material model. Thismodel has been created as an extension to a previously de-veloped model which used a linear viscoelastic material tomodel brain white matter (Abolfathi et al., in press). Materialproperties of the constituents and the geometry of the com-posite are taken as input data and the model imposes period-icity constraints on a RUC as different loading scenarios areapplied to it. The homogenized response of the composite,the average stresses carried within each constituent of thecomposite, and the maximum local stresses are obtained. Foreach of the loading scenarios, the impact of geometrical vari-ables such as the axonal fiber volume fraction (Vf ) and undu-lation ratio of the axons (U) are evaluated. The purpose of thismodel is to gain a better understanding of the microscopicstresses and strains in a unit cell composed of axonal fibersand extracellular matrix under loading. The model can alsobe used to better understand how macroscopic stresses aretransferred to the microstructure of brain tissue. The tissuesbeing modeled in this example are the human brainstem andheterogeneous white matter in the brain.

2. Hyperelastic theory and anisotropic mate-rial models

A hyperelastic material constitutive model is used to definethe response of the materials in this model. Hyperelasticmaterials use stress–strain relationships that are derivedfrom strain energy density functions (Bhatti, 2006). Thehyperelastic constitutive model is used to describe materialsthat are able to undergo large, recoverable elastic strainsuch as rubber-like polymers and soft biological tissues.An isotropic material has the same mechanical responseregardless of loading direction. For an isotropic hyperelasticmaterial the strain energy density function W is a scalarfunction of the right Cauchy–Green deformation tensor, C.The scalar function is composed of either the principalinvariants or the principal stretches of the deformation,both of which are derived from the right Cauchy–Greendeformation tensor. The right Cauchy–Green deformationtensor, is obtained from the deformation gradient, F and isgiven by

C = FT : F, F = ∂x/∂X. (1)

J O U R N A L O F T H E M E C H A N I C A L B E H AV I O R O F B I O M E D I C A L M A T E R I A L S 2 ( 2 0 0 9 ) 2 4 3 – 2 5 4 245

The deformation gradient relates the position of the materialin the initial configuration, X, to its position in the deformedconfiguration, x. The principal invariants of C are

I1 = trC, I2 =12[(trC)2 − tr(C : C)], I3 = detC. (2)

The corresponding strain energy density function in terms ofthe principal invariants would be of the form,

W =W(I1, I2, I3). (3)

The principal invariants can also be obtained from theprincipal stretches through the following relationships:

I1 = λ21 + λ

22 + λ

23, I2 = I3(λ

−21 + λ

−22 + λ

−23 ), I3 = λ

21λ

22λ

23. (4)

The principal stretch, λi (i = 1, 2, 3), is defined as

λi =√ei ·U : U · ei, (5)

where U2= FT : F and ei is a principal direction of U. A strain

energy density function in the terms of principal stretcheswould be in the form,

W =W(λ1, λ2, λ3). (6)

Either of the forms presented in (3) or (6) can be used todescribe the hyperelastic response of an isotropic material.Strain energy functions can used to describe the elasticresponse of compressible and incompressible materials.The volume of a compressible material can change duringdeformation while it will remain constant within anincompressible material. Most hyperelastic materials areeither incompressible or very close to incompressible andbrain tissue is considered to be very close to incompressible.

In this work the Ogden hyperelastic formulation has beenused to describe the tissue. Soft biological tissue is proven tobe represented well by the Ogden formulation andmost of themechanical test data available for brain tissue in literature fitswith an Ogden hyperelastic function; see Coats and Margulies(2006), Miller and Chinzei (2002), Miller et al. (2000), Prangeand Margulies (2002), and Velardi et al. (2006). The Ogdenhyperelastic function is given by

W =N∑i=1

2µiα2i

(λ̄αi1 + λ̄

αi2 + λ̄

αi3 − 3)+

N∑i=1

1Di(Jel − 1)2i. (7)

In this equation, µi is the shear modulus of the material,while αi andDi are the remainingmaterial parameters neededto fully define the material.

The aforementioned hyperelastic material models areisotropic and the strengthening effect of fibers in a compositecan be added to the model through the addition of pseudoinvariants, as seen in Spencer (1984). These additionalterms to the isotropic strain energy function are usedto represent the behavior of fiber-reinforced transverselyisotropic composites. A unit vector, a0, shows the fiberorientation and the additional invariants are defined asfollows:

I4 = a0 · C · a0, I5 = a0 · C : C · a0. (8)

The strain energy function can be split into several partswhere the first part represents the material’s isotropic

response and the remaining parts represent the directionalproperties provided by the fibers:

W =W(I1, I2, I3)+W(I4)+W(I5). (9)

Strain energy density functions in the above form are usedto add the energy contributed from the matrix and the fiberof the composite as seen in Qui and Pence (1997), Quapp andWeiss (1998), Holzapfel (2000), and Merodio and Ogden (2003,2005). The fiber–matrix interaction also contributes to theanisotropy of the material and its effect has been consideredby Peng et al. (2006) and Guo et al. (2006). They consider this bydecomposing the deformation gradient into two parts. One ofthe parts is for the uniaxial deformation in the fiber directionand the other part is for the remaining shear deformations.An Ogden hyperelastic model that was augmented with theadditional invariants was used to describe the mechanicalbehavior of brain tissue by Merodio and Ogden (2003). Thismodel was further verified by experiments performed byVelardi et al. (2006).

The constitutive equations that are derived from strainenergy density functions, describe the material in amacroscopic nature as a continuum mainly in the formof a mathematical equation that fits to the experimentaltest results. The approach does not give any informationabout the mechanisms of deformation or the behavior ofthe microscopic structure of the material (Holzapfel, 2000).The micromechanics model presented here is attempting toprovide a link between the macroscopic and the microscopicbehavior of the composite structure. The necessity forthis link has been discussed by Meaney (2003). Each ofthe constituents is defined with an isotropic hyperelasticmodel of a different stiffness; thus creating an anisotropiccomposite. The anisotropy that is added by the axonalfiber–matrix interface is all accounted for within the model.The model can also predict the full anisotropic behavior ofa composite, allowing undulation to be studied whereas thephenomenological formulations are limited to isotropic ortransversely isotropic materials.

3. Micromechanics finite element model

The micromechanical method characterizes composite be-havior from the known properties of constituents throughthe modeling and analysis of a representative volume ele-ment. In this process the heterogeneous composite can bereplaced by a homogeneousmedium. Micromechanics princi-ples are being practiced in composite engineering with greatsuccess. Such principles can also benefit from enhancementof the characterization methods in tissue engineering. InAbolfathi et al. (in press), the authors introduced the mi-cromechanical modeling concept while simulating brain tis-sue. The procedure presented in this paper extends the samemicromechanical modeling method which utilizes finite ele-ment methods (FEM) in the analysis of soft biological tissuesundergoing large deformation. The method allows one to pre-dict the response of a composite material when the materialproperties of the constituents within the composite and thegeometrical arrangement of the constituents are known. Inthis paper, the material behavior is confined to hyperelasticand under large deformation. Inclusion of the viscoelastic na-ture of the tissue, in addition to the hyperelastic and large de-formation will enhance the solution in many circumstances.


Fig. 1 – (a) Histology slide showing the axon distribution in the cross-section of an adult porcine brainstem (Arbogast andMargulies, 1999), (b) periodic microstructure of a simulated hexagonal distribution of axons inside an extracellular matrix(ECM), and (c) an RUC with a hexagonal distribution of axons.

3.1. Micromechanical unit cell modeling

A histology slide representing the axonal distribution insidethe brainstem as shown in Fig. 1(a) is used. It was observedthat the conventional hexagonal packing considered in thecomposite mechanics can be used to represent the repetitivedistribution of axons inside the extracellular matrix. Thus,this study is utilizing the hexagonal distribution of axons todevelop different composite tissue RUCs; see Fig. 1(b). Thevolume fraction of axons in the composite tissue models isconsidered to be 53%, a value that was reported in a studyby Arbogast and Margulies (1999). However, different volumefractions of axons are considered here to study their effect onthe overall behavior of the composite tissue.

Axonal distribution within brainstem areas as observed byArbogast and Margulies (1999) is highly oriented. Also in thecorpus callosum area, similar uniaxial distributions for axonsare observed and hence both are modeled as unidirectionalcomposite tissues. To represent regions other than thebrainstem and corpus callosum, composite tissue modelswith undulated axons are created. The undulated domainshown in Fig. 2 is selected from a guinea-pig optic nerve(Meaney, 2003). There is no real experimental data availableon the waviness geometry of axons. Hence, the maximumundulation ratio reported by Meaney (2003) is utilized todefine the undulation ratio range with the minimum value ofone corresponding to a straight axon. The undulation ratio isdefined as the ratio of the length of the sinusoidal curve (lt) tothe wavelength (lo) (Fig. 3). Based on this range, three differentundulation ratios are considered to develop different wavymodels (Table 1). The undulation is considered to determineits effect on the behavior of the composite tissue. Fig. 4(a)and (b) show the RUCs for composite tissue representing thestraight and wavy distribution of the axonal fibers.

A unit cell with the geometrical cross section given inFig. 2 is chosen. The length of the unit cell must somehowcover one full periodic length of the axon. The RUC under anyloading scenario can be analyzed. The outcome will be thestresses and deformations found by using the ABAQUS finite

Table 1 – The undulation ratios that were used alongwith the corresponding amplitudes and periods(non-dimesionalized with axon diameter) that wereused to define the sinusoidal shape of the RUCs(Meaney, 2003)

Undulation ratio Amplitude Period

U1 1.000 0 340U2 1.131 41 340U3 1.262 60 340

element package (ABAQUS, 2006) interfaced with the periodicboundary conditions subroutines developed by the authors.The stress–strain information obtained from the analysis arevolume-averaged over the volume of the RUC, i.e.

σ̄ij =1V

∫vσijdV, ε̄ij =

1V

∫vεijdV (10)

where σ̄ij and ε̄ij are the volume-averaged stresses and strains,and σij and εij are the distributed stresses and strains,respectively. V is the volume of the RUC.

3.2. Loading and periodicity constraint

The unit cells are subjected to three sets of loadings, twonormal and one shear. The details of these load cases canbe seen in Fig. 5. In load case 1, the load is applied in thedirection of the longitudinal axis of the axon, while in loadcase 2, it is applied perpendicular to the direction of the axon.Each load case is obtained by applying a finite displacementto opposite sides of the cell. Stresses are recorded for eachelement and are further used to calculate volume-averagedstresses. These volume-averaged stresses are plotted againstthe applied stretches to show the overall material behavior.

Periodicity constraints: Periodicity requires that oppositefaces of the unit cell deform identically. This requires certainconstraint relations between the nodes on the faces. Invokingthese constraints requires the number and distribution ofnodes on opposite faces to be identical. It is also convenientto have a node located at the geometric center of each face.


Fig. 2 – (a) The distribution of undulated axons inside the extracellular matrix for a guinea-pig optic nerve microstructure(Meaney, 2003), (b) simulated periodic sinusoidal wavy distribution of axons, with the corresponding periodic unit cell, and(c) RUC representing wavy periodic microstructure of the composite tissue (d = fiber diameter, w = width of unit cell,h: height of the unit cell =

√3).

Fig. 3 – (a) Schematic representation of the undulationratio, U of a sinusoidal wave with amplitude, A. Theundulation size is defined as the ratio of the length of thesinusoidal curve (lt) to the wavelength (lo).

To show how these constraints are achieved, consider again

the solid model shown in Fig. 6 (Garnich and Karami, 2004;

Karami and Garnich, 2005; Naik et al., 2008). On this geometry

there are 6 faces, 12 edges, 6 center-face nodes and 8 corner

nodes. The displacement degrees of freedom for the nodes on

half of the faces (2, 4 and 6), edges (24,26,46, . . .) and corners

(246, . . .)must be written in terms of the degrees of freedom of

the nodes on the other half. These algebraic relations are such

that they force opposite faces to deform to the same shape

though they may have a rigid body translation between them.

To enforce the repeating behavior, the following constraintrelations are enforced. In the following, ui (i = 1,2,3)represents the displacement in the ith-direction, ci (i =1,2, . . . ,6) represents the center face nodes, ni and njrepresent the node pair on opposing faces i and j, eij stands forthe edge ij, sharing the faces i and j,nij stands for the nodeslocated on edge eij, and finally nijk represents the corner nodesharing faces i, j and k. The constraint equations are definedsuch that displacement components of each node on faces 2,4 and 6 are removed in terms of the respective componentsfor the pair node on faces 1, 3 and 5. To enforce deformationcompatibility between opposite faces yet still allow a rigidbody motion between the two faces, the displacements forthe nodes on each face are expressed relative to the centernode on that face. Because edge and corner nodes areshared between multiple faces, care must be taken to avoidredundant (over) constraints.

Additional constraints on the center nodes of the oppositefaces are applied. The slave nodes on the center of faces 2, 4and 6 are related to the active nodes on the faces 1, 3 and 5 asshown below:

uc2i = −u

c1i , u

c4i = −u

c3i , u

c6i = −u

c5i (i = 1,2,3). (11)

Node pairs on faces 1 and 2For all nodes on face 2 except the center node and the

nodes on edges e24 and e26 and the corner node n246, one


Fig. 4 – (a) Repeated unit cell (RUC) of axon and matrix with axon undulation. (b) assumed hexagonal distribution of theaxonal fibers within a RUC with straight axons (zero undulation).

Fig. 5 – Illustrations of the different load cases showing the directions that displacement is applied. Load cases 1 and 2produces pure axial stresses in directions 1 and 2, and load case 3 produces pure shear if the material is completelyisotropic.

has,

un2i = u

n1i − u

c1i + u

c2i . (12)

Applying Eq. (8) one gets,

un2i − u

n1i − 2uc1i = 0. (13)

On the edges e24 and e26, the following constraint relationsapply respectively,

un24i = u

n13i − 2(uc1i + u

c3i )

un26i = u

n15i − 2(uc1i + u

c5i ).

(14)

At corner node n246, one has

un246i = u

n135i − 2(uc1i + u

c3i + u

c5i ). (15)

Node pairs on faces 3 and 4For all nodes on face 4 except the center node, the corner

node n246 and the nodes on edges e24 and e46, one has;

un4i = u

n3i − 2u

c3i . (16)

For the nodes on edge e46 the following relation is enforced.

un46i = u

n35i − 2(u

c3i + u

c5i ). (17)

Node pairs on faces 5 and 6For all nodes on face 6 except the center node, the corner

node n246, and the nodes on edges e26 and e46 one has;

un6i = u

n5i − 2u

c5i . (18)

In addition to the periodic constraints enforced on the node-pairs on opposite surfaces of the unit cell, translational androtational constraints to prevent the singularity of rigid bodymodes are also enforced.

Rigid body constraints: The central node at the center ofthe body is fixed in all directions. To prevent translationsand rotations, the center-nodes on faces 1 and 2 are fixed indirections 2 and 3, and finally a node on one of the edges offace 1 is fixed in direction 2 or 3, depending on the edge, toprevent rigid rotation about the length of the model.

3.3. Material input for axon and matrix

To develop the micromechanical model to represent thecomposite tissue behavior, it is essential to have propermaterial characteristics of the constituents. The materialproperties for axons are obtained from an experimental study


Fig. 6 – (a) RUC representing the hexagonal distribution microstructure of the composite material, and (b) RUC representingthe wavy microstructure of composite material with hexagonal packing assumption.

by Meaney (2003) on guinea-pig optic nerves. Since thereis a resemblance between the organic architecture of thehuman optic nerve and the guinea-pig optic nerve (Waxmanet al., 1995; Bain et al., 2003). The reported data by Meaneyis utilized in the present study. The volume fractions ofaxons inside the optic nerves are observed to be higher than90% (see; Meaney (2003) and Arbogast and Margulies (1999)).This means that there should be only a minute or negligibledifference between the properties of axons and the compositetissue and, hence, the optic nerve material properties canbe used to represent the material for axons. A similarassumption was also considered by Arbogast and Margulies(1999) while developing analytical models to represent thehuman brainstem.

In the Meaney (2003) study, data for the optic nervetissue was reported with Ogden formulations and the sameformulation was utilized in this method. The axons areobserved to be three times stiffer than their surroundingmatrix (Arbogast and Margulies, 1999). The Ogden parameter,µi, is the shear modulus of the material and its value forthe axon is provided by the optic nerve tissue data (Meaney,2003). For the matrix, it is derived by dividing the respectivevalue for the axons by a factor of three, whereas the othertwo parameters (Di & αi) are kept the same and the materialparameters of the constituents are given in Table 2.

Table 2 – The coefficients that were used to define theOgden hyperelastic material for the axons and thematrix (Meaney, 2003)

µ (Pa) α D (Pa−1)

Axon property 290.82 6.19 1.00E−08Matrix property 96.94 6.19 1.00E−08

4. Numerical stress analysis of undulatedaxons

Numerical results obtained from the studies include thevolume-averaged stresses for the RUC, volume-averagedstresses within each constituent, contour stress plots, andmaximum stresses at the applied displacements for eachof the loading scenarios. The volume-averaged stressesof the composite provide information about the overallmechanical response and they are useful for comparison withexperimental results that describe the macroscopic behaviorof brain tissue. The contour stress plots, volume-averagedstresses within the constituents, and the maximum stressesare valuable because they describe what is happening in thetissue at the microscopic level. There is a limited amount ofdata available for defining the geometry of whitematter in thebrain so the impact of the geometrical variables was studied


Fig. 7 – The change in undulation as a stretch is applied inthe axon direction is compared with experimental resultsand the model created by Bain et al. (2003).

with the model. The impact of the undulation ratio (U) wasstudied by evaluating three RUCs with different values of U ata fixed fiber volume fraction Vf . To determine the impact ofVf , it was varied on several RUCs with no undulation (U = 1).

4.1. Axon undulation change with stretch

Numerical results for undulation change with stretch wasfirst compared with the experimental results of Bain et al.(2003) in Fig. 7. This comparison provides a good picture ofhow the results of this model compare with the availablemacroscale data. The comparison clearly shows that thereis work which still needs to be done in refining this modelas well as experimental setups to gather microscale data.Much of the inaccuracy illustrated in these comparisonsstems from the fact that there is limited data available forthe geometry and constituent material properties of braintissue. The change in axon undulation was experimentallymeasured at different levels of tissue strain by Bain et al.(2003) and they also created three differently based modelswith different levels of coupling between the axon and thematrix. Fig. 7 shows the axon undulation at different levelsof tissue strain for their experimental results and theirmicrostructural model for complete coupling when the axonhas an initial undulation of 1.131 and the maximum stretchapplied was 1.25. Bain et al. ’s results are compared withresults generated by this model and it is seen that this modelis slightly better than their microstructural model, but it isstill not a good match with the experimental results. In Bainet al. (2003) it was found that a microstructural model witha gradual coupling between the axon and matrix fits theexperimental data best. It is necessary to define an adhesionbetween the axon and matrix instead of the perfect bondingthat is currently being assumed.

4.2. Stress distribution of undulated axons

Experimental tensile test results for the corpus callosumwere published by Velardi et al. (2006). The corpus callosumis composed of bundles of axonal fibers that are highly

Fig. 8 – The response of the composite with differentvalues of Vf to a uniaxial stretch in the axon direction.Stresses are given in terms of Lagrange stresses (basedupon undeformed area) and compared with experimentalresults for the corpus callosum from Velardi et al. (2006).

uniaxially oriented and the load was applied in thelongitudinal direction of the axons up to a stretch of 1.25. Theresults of this test are shown with vertical lines representingthe standard deviation along with results generated by themodel, being presented in Fig. 8. The stresses which aregraphed from the model are Lagrange stresses because thatis also how the experimental results are being presented.The axon volume fraction is not known for the tissue samplethat was tested, so results from several volume fractionsare plotted. The model does not fit the data very well atintermediate strains and the α term may need to be adjustedto create a better fit.

Models were created with undulation ratios of U1, U2,and U3 and were then tested by applying a uniaxial stretchin the longitudinal direction of the axon. The RUCs werecompressed to a stretch of 0.5 and extended to a stretch of 1.5in a number of incremental steps. The geometrical variable ofinterest in this portion of the study was the undulation of theaxons, which was studied at constant Vf of 53%.

Volume-averaged stresses within each constituent werecalculated for the different undulation ratios at differentstretch levels. These results can be seen in Fig. 9 and it showsthat the level of axon undulation has a much greater impacton these results. Stress, σ11, is shown in both the axon andthe matrix for each of the different initial undulations. Whenthere is zero undulation (U1), the axon and matrix stressesdiffer by a large amount, with the stress in the axon threetimes greater than the stress in the matrix at a stretch of1.5. As the undulation increases, the stresses become moreequally shared between the axon and the matrix, and finally,at U3 the stress in the axon is only 1.6 times greater thanthe stress in the matrix. In the axon’s longitudinal direction,the axon stress decreases while the matrix stress increasesas the undulation of the axons increases. The distribution ofstresses in the constituents can also be examined by lookingat contour stress plots such as the ones shown in Fig. 10. Thisplot shows a stress contour plot of σ11 for U3 at a stretch of1.5 and the composite’s constituents are separated making iteasier to distinguish between them. From these results one


can conclude that if an injury criterion is being examined forthe axons it would be important that undulation be taken intoconsideration.

The maximum stresses in each direction can be obtainedfrom the contour plots for each U at different levels ofstretch. These results are presented in graphs of stress versusstretch for σ11 in Fig. 11 and also for σ22 and σ12 in Fig. 12.The maximum stresses in U2 and U3 are greater than themaximum stresses in U1 where there is no undulation. Themaximum stress for σ11 at a stretch of 1.5 in U3 is 14% greaterthan in U1 and for a stretch of 0.5 the stress in U3 is 31%greater than in U1. The differences between the maximumstress in U2 and U3 are much less than the differencesbetween U1 and either of the other undulations. From thisobservation one may conclude that the exact magnitude ofundulation is not as important as determining whether or notthere is any undulation. The plots of σ22 and σ12 in Fig. 12show that stresses in these directions develop for U2 and U3,but not for U1. These stresses are much larger in magnitudefor compressive displacement loads than they are for tensileloads and σ22 shows this behavior most prevalently.

Fig. 9 – The volume-averaged normal stresses in direction1 (along the direction of straight axons) in each of theconstituents for different values of undulation as a uniaxialstretch is applied in the axon direction (load case 1).

The axon undulation did not have a large effect on thevolume-averaged stress of the composite, but it did have

Fig. 10 – Contour stress plots showing the tensile stresses in an undulated RUC that is stretched in the direction of thelongitudinal axis of the axon. The composite (a), the axon (b), and the matrix (c) are shown for U3 at a stretch of 1.5.


Fig. 11 – Maximum normal stresses in the direction ofloading (σ11) for different values of undulation as a uni-axial stretch is applied in the axon direction (load case 1).

Fig. 12 – The developed maximum normal stresses (σ22) inthe direction perpendicular to axons and maximum shearstress (σ12) for different values of undulation as a uniaxialstretch is applied in the axon direction (load case 1).

a large effect on how the stresses were distributed amongthe constituents and also on the maximum stress values.Furthermore, the results from the two different undulations(U2 and U3) were similar,leading one to believe that theactual magnitude of undulation may not be as critical asdetermining whether or not there is undulation present.

4.3. Effect of axon volume fraction

The second geometrical variable of interest was the axonvolume fraction, Vf . This variable was studied by applyingdifferent loads to RUCs with no undulation and with Vf equalto 40%, 53%, and 60%. The loads were created by applyingdisplacements in the longitudinal direction of the axon (loadcase 1), a direction perpendicular to the longitudinal directionof the axon (load case 2), and a shear displacement along thelongitudinal direction of the axon (load case 3). The volume-averaged stresses of the composite, the volume-averagedstresses in the constituents, the stress contour plots, and themaximum stresses can all be obtained for each of the abovedescribed scenarios, but only a few of these results will bepresented here to display the capabilities of the model.

It was found that the stresses for higher values of Vf arelarger for a given displacement because there is a greater

Fig. 13 – The developed stresses in the tissue (made ofaxon and matrix) in the direction perpendicular to the axonin response to a uniaxial stretch applied (load case 2) atdifferent values of Vf .

percentage of the stiffer axons present in these models. This

result was expected. At a stretch of 1.5, σ11 for Vf = 60% is 22%

greater than it is for Vf = 40%. The same relation can seen for

a compressive stretch value of 0.5 where σ11 for Vf = 60%

is 22% greater than it is for Vf = 40%. These differences are

substantial and the same can also be said for load case 2when

the load is applied in direction 2. Fig. 13 shows σ22 as this

is the stress that is created when displacement is applied in

direction 2. For load case 2, σ22 for Vf = 60% is 16% greater

than it is for Vf = 40% at a stretch of 1.5. For a stretch of

0.5, the stress σ22 with Vf = 60% is 26% greater than it is for

Vf = 40%.

An interesting result found when comparing the stresses

created from load case 1 and load case 2 is shown in Fig. 14.

The curves in this figure can be used to compare the volume-

averaged stresses in the direction of loading for the axon and

matrix during load case 1 and load case 2. The curves show

that the stresses are more equally distributed between the

axon and thematrix for load case 2 than they are for load case

1. In load case 1, the axon stress is 3 times greater than the

matrix stress, while in load case 2 the axon stress is only 1.2

times greater than the matrix stress. The stresses are more

equally distributed between the constituents for load case 2

resulting in a lower stress in the axon and a greater stress in

the matrix as compared to load case 1.

Shear stresses were created by applying a shear displace-

ment to the RUCs and the effect of the Vf was examined. The

displacement is applied in the direction of the axon’s longi-

tudinal axis and the stress created is σ12. Fig. 15 shows the

maximum shear stresses for shear strains from 0 to 0.5. The

axon is a stiffer material than the matrix so for higher values

of Vf , the composite is also stiffer and the stresses at each

strain are higher. At a shear strain of 0.5, the volume-averaged

shear stress for Vf = 60% is 25% higher than it is for Vf = 40%.

The maximum shear stresses followed the same trend and at

a shear strain of 0.5 the maximum shear stress for Vf = 60%


Fig. 14 – The developed volume-averaged normal stressesin each of the constituents (axon and matrix) in thedirection of loading (σ11, and σ22) for each of load cases 1and 2 with Vf = 53%.

was 18% greater than it was for Vf = 40%. The stresses are dis-tributed throughout the axon andmatrix and this distributioncan be seen in Fig. 16. The load applied for the results in thisfigure was a shear strain of 0.5 on a RUC with Vf = 53% andno axon undulation. In summary, the axon volume fraction,Vf , had a recognizable impact on all of the results because thecomposite becomes stiffer as the volume fraction increases.

5. Conclusion

A three-dimensional micromechanics model for brainwhite matter tissue comprised of axons and matrix waspresented. The behavior of a periodic unit cell representinga microstructural volume element of a well-organizedheterogeneous tissue was examined under axial as well asshear loadings. To study the impact of axon waviness andundulation, straight as well as undulated axons were studied.Parametrical studies on the mechanical behavior of the tissueunder such conditions were conducted. It was found that

Fig. 15 – The developed maximum shear stresses in thetissue under shear loading (load case 3) as shear strains areapplied for different values of Vf .

as the axon undulation increased, the maximum stress and

the stress in the matrix increased while stress in the axons

decreased. The axon volume fraction was found to have an

impact on the stress magnitudes as higher axon volume

fractions lead to higher stresses both in the composite and in

the constituents. When the kinematical load was applied in a

direction perpendicular to longitudinal axis of the axon, the

stresses in the axon and matrix were more evenly distributed

than they were for a displacement along the longitudinal axis.

In comparing the results of this model with the limited

experimental data available, it was concluded that, although

the agreement is satisfactory, work is still needed to refine

the model to include enhanced microstructural details of the

geometry, as well as the material property of the constituents

of the brain tissue. In general, for a complete biomechanics

characterization and analysis of brain tissue at microscale

there is still a long road ahead. This model can serve as a firm

foundation for microstructural analysis once more accurate

experimental information on the microscale geometry and

the mechanical characteristics becomes available.

Fig. 16 – Contour plots showing the shear stress in an initially straight RUC with a shear strain applied. The Vf = 53% andshear strain = 0.5. Contour plots show the composite tissue (a), the axon (b), and the matrix (c).


Acknowledgement

The authors would like to acknowledge the Air Force Officeof Scientific Research (AFOSR) for the financial support of thiswork.

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A micromechanical hyperelastic modeling of brain white matter under large deformation