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Research Paper

Exploring the mechanical behavior of degradingswine neural tissue at low strain rates via thefractional Zener constitutive model

Sarah A. Bentiln, Rebecca B. Dupaix

Department of Mechanical and Aerospace Engineering, The Ohio State University, 201 W. 19th Avenue,Columbus 43210, OH, USA

a r t i c l e i n f o

Article history:

Received 18 April 2013

Received in revised form

19 October 2013

Accepted 24 October 2013

Available online 7 November 2013

Keywords:

Brain tissue

Degradation

Viscoelasticity

Constitutive model

Fractional Zener

ANOVA

nt matter & 2013 Elsevie.1016/j.jmbbm.2013.10.020

hor. Tel.: þ1 614 292 8404: bentil.1@buckeyemail.os

a b s t r a c t

The ability of the fractional Zener constitutive model to predict the behavior of postmortem

swine brain tissue was examined in this work. Understanding tissue behavior attributed to

degradation is invaluable in many fields such as the forensic sciences or cases where only

cadaveric tissue is available. To understand how material properties change with

postmortem age, the fractional Zener model was considered as it includes parameters to

describe brain stiffness and also the parameter α, which quantifies the viscoelasticity of a

material. The relationship between the viscoelasticity described by α and tissue degrada-

tion was examined by fitting the model to data collected in a previous study (Bentil, 2013).

This previous study subjected swine neural tissue to in vitro unconfined compression tests

using four postmortem age groups (o6 h, 24 h, 3 days, and 1 week). All samples were

compressed to a strain level of 10% using two compressive rates: 1 mm/min and 5 mm/

min. Statistical analysis was used as a tool to study the influence of the fractional Zener

constants on factors such as tissue degradation and compressive rate. Application of the

fractional Zener constitutive model to the experimental data showed that swine neural

tissue becomes less stiff with increased postmortem age. The fractional Zener model was

also able to capture the nonlinear viscoelastic features of the brain tissue at low strain

rates. The results showed that the parameter α was better correlated with compressive rate

than with postmortem age.

& 2013 Elsevier Ltd. All rights reserved.

r Ltd. All rights reserved.

; fax: þ1 614 292 3163.u.edu (S.A. Bentil), dupaix.1@osu.edu (R.B. Dupaix).

1. Introduction

Characterization of the material response of brain tissue isinvaluable in medical applications such as robotic surgery,where prediction of brain deformation is needed; surgicaloperation planning and surgeon training systems, whereforce feedback is needed; and traumatic brain injuries, where

injury mechanisms are needed to develop predictive modelsfor preventative purposes (Mac Donald et al., 2011; Miller andChinzei, 1997; Wang et al., 2006). The mechanical behavior ofbrain tissue extracted following dynamic and quasi-staticexperimental tests using different animals and deformationmodes can vary between samples, despite using identicalexperimental conditions (Bilston et al., 1997; Cheng et al.,

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2008; Estes and McElhane, 1970; Gefen and Margulies, 2004;Miller and Chinzei, 1997; Prange and Margulies, 2002). Someof the variability is attributed to temperature, sample pre-paration, and postmortem age of the tissue (Bilston, 2011; Ranget al., 2001).

Constitutive models such as hyperelastic, poroelastic, andviscoelastic have been developed to describe the response ofneural tissue subjected to unconfined compression tests atlow strain rates, which are typical for medical applications(Cheng and Bilston, 2007; Davis et al., 2006; Miller andChinzei, 1997). These models are not designed to predictbrain tissue degradation, but have been used in the past tocapture the behavior of degrading tissue by re-fitting arbitraryconstants at the different postmortem times. A fractionalcalculus based model may therefore be ideal for exploringthe relationship between viscoelasticity and tissue degrada-tion. Fractional calculus is a branch of mathematics that wasdeveloped around the same time as classical, integer basedcalculus (Kulish and Lage, 2002). The fractional calculustechnique has served as a tool in science and engineeringcovering topics such as fluid flow, rheology, diffusive trans-port, electrical networks, electromagnetic theory, and prob-ability (Kulish and Lage, 2002; Soczkiewicz, 2002). It has beenused to describe soft biological tissues, including brain, butthe fractional calculus parameters has not been consideredas a metric for predicting tissue degradation (Craiem andArmentano, 2007; Davis et al., 2006; Franceschini, 2006; Klattet al., 2007; Kohandel et al., 2005; Libertiaux and Pascon, 2009;Montgomery, 2009).

The purpose of this study is to illustrate the power of thefractional Zener (FZ) constitutive model, derived using thefractional calculus approach, to explore tissue responsechanges due to degradation. Unlike integer based viscoelasticmodels, the FZ model contains a parameter (α), whichquantifies the viscoelasticity of a material by determiningits location on a viscoelasticity spectrum. The value of α

allows the FZ model to vary between a material that exhibitssolid-like behavior to fluid-like behavior. We hypothesize thatα may work well at describing the interaction between theviscoelasticity of the material with postmortem age. Viscoe-lastic parameters such as moduli and relaxation time canalso be obtained using the FZ model, as with other integerbased viscoelastic models.

Table 1 – Total number of samples tested for each postmortem

Postmortem time

o6 h

HemisphereLeft 21Right 40

LobeRegion A (�occipital) 20Region B (�temporal) 21Region C (�frontal) 20

Compressive rate1 mm/min 315 mm/min 30

2. Method and theory

2.1. Data acquisition

Swine brains from a local abattoir were subjected to uncon-fined compression tests in a previous study (Bentil, 2013).These porcine heads were from swine that were between theages of 6 and 10 months, with an average weight of 250 lbs.Porcine brains were extracted from the skulls immediatelyafter the animals were harvested, and the dura mater wasremoved. Tissue samples were prepared by first sectioningthe left and right hemispheres of the cerebrum using ascalpel. The cerebrum is a heterogeneous region of the brainthat consists of both gray and white matter. A cylindricalcoring tool was used to generate brain tissue samples fromthe cerebrum. A scalpel was used to generate planar endsurfaces, which ensured that these two surfaces weresmoothed to reduce any tissue stress non-uniformity duringdata collection. The average height of all the tissue sampleswas 12 mm. The diameter of all tissue samples was 15 mm tofacilitate extraction of three samples from each hemisphere,as opposed to the two 30 mm diameter samples per hemi-sphere obtained by Miller and Chinzei (1997).

Four postmortem ages (o6 h, 24 h, 3 days, and 1 week) andsix regions of the brain were considered. These regionscomprised of samples extracted from the occipital, temporal,or frontal lobe of either the left or right hemisphere. Table 1categorizes the total number of samples tested by hemi-sphere, lobe, and compressive rate. Tissues were stored inphysiological saline solution placed in a 5 1C refrigerator untiltesting. All samples were tested at room temperature.

The unconfined compression tests of the porcine cere-brum samples were performed, using the RSA III (Rheo-metrics Solids Analyzer). The load cell capacity for the RSAIII was 350 gf (�3.4 N). Parallel cylindrical plate tool geometrywas used with the test device. Testing began immediatelyafter the top plate visually appeared to be in contact with thetop surface of the tissue sample. Contact was also confirmedby a non-zero force readout on the RSA III. During the test,the tissue samples were subjected to a ramp-hold straininput. This input controls the movement of the cylindrical topplaten on the tissue. Compressive rates of 1 and 5 mm/min

time cohort by hemisphere, lobe, and compressive rate.

24 h 3 Days 1 Week

25 22 2351 38 29

28 20 1624 22 1924 18 17

35 28 2941 32 23

(α, E3, τ)

E1

E2

σ ε

Fig. 1 – Fractional Zener constitutive model.

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were used for the ramp segment of the strain input. Thehold region began when the brain was compressed to 10%strain and ended following a duration of 20 s. Twenty sec-onds duration provided enough data points to generate thetrend attributed to tissue relaxation effects. Displacementand force data was acquired using the TA InstrumentsThermal and Rheological data analysis software. This datawas then used to calculate the engineering strain and stress,from which material properties of the brain tissue can bedetermined. All samples were tested once. Additional detailsof the sample preparation and test protocol can be found inBentil (2013).

2.2. Fractional calculus

The benefit of using fractional calculus to describe viscoelas-tic materials is apparent when the one-dimensional (1D)constitutive model is used as an example to describe thestress–strain relationship. The fractional 1D constitutivemodel relies on the order of the derivative for the strain tocharacterize the viscoelastic behavior of a material. Consti-tutive equations for rheological elements can be describedusing the fractional derivatives and integrals (differintegral)operator, aD

αt f ðtÞ, defined as (Podlubny, 1999)

Dαf ðtÞ ¼ dαf ðtÞdtα

ð1Þ

In this notation, aDαt f ðtÞ represents the differintegration of a

function f ðtÞ to any arbitrary order α. The superscript α is areal number that enables the differintegral operator to beused in differentiation when α is positive and integrationwhen α is negative. The lower and upper limits of integrationare denoted by the subscripts α and t, respectively (Podlubny,1999; Ross, 1977). Since strain is the function of interest,f ðtÞ ¼ εðtÞ. Constitutive equations of rheological models (i.e.,springs, and dashpots) can be described in general form usinga fractional element.

The differintegral operator, when used to describe theconstitutive differential equations of rheological models,yields the fractional element. The fractional element(Eq. (2)), which has an order (α) of strain between 0 and 1describes the linear viscoelastic region of a viscoelastic mate-rial (Podlubny, 1999). Traditional rheological elements, such assprings and dashpots, can be derived from the fractionalelement by setting α to 0 or 1. A zero order derivative of strainrepresents a Hookean solid (Eq. (3)) and order one describes aNewtonian fluid (Eq. (4)) (Meral et al., 2009; Podlubny, 1999).

Fractional ð0oαo1Þ sðtÞ ¼ EταDαεðtÞ ð2Þ

Spring ðα¼ 0Þ sðtÞ ¼ Eτ0D0εðtÞ ¼ EεðtÞ ð3Þ

Dashpot ðα¼ 1Þ sðtÞ ¼ Eτ1D1εðtÞ ¼ η_εðtÞ ð4Þwhere E is the elastic property of the material (pressure units),τ is the relaxation time (time units), η is the viscocity of thematerial (pressure� time units).

2.3. Fractional Zener constitutive model

The mechanical representation of the fractional Zener model(Fig. 1) is created using two springs and a fractional element.

Setting the parameter α to 1 yields the traditional Zenermodel, also referred to as the standard linear solid (SLS)model. Solving the system of equations for the differential FZmodel (Fig. 1) yields the stress response of the brain tissueusing four parameters E1, E0, τ0, and α

sðtÞ þ τα0DαsðtÞ ¼ E1εðtÞ þ E0τ

α0D

αεðtÞ ð5Þ

where E1 ¼ E1,E0 ¼ E1 þ E2,τ0α ¼ E3=E2

� �τα.

The brain stiffness can be described by the parameters E1and E0. The relaxation time is described by τ0, while α gives asense of the brain material location on the viscoelastic spectrum.

The three most common differintegral operator definitionsused to solve fractional derivative and integral equations arethe Riemann–Liouville (RL), Grünwald–Letnikov (GL), andM. Caputo (Das, 2007). Rather than solving the fractionaldifferential equation (Eq. (5)) using these common definitions,the FZ constitutive equation was solved by first transformingthis equation into the Laplace domain, solving the simplertransformed algebraic problem, then applying an inverseLaplace transform to obtain the solution. A nonlinear leastsquares algorithm was then applied using MATLAB (MATLAB,2008) to determine the values of the four parameters that willyield the stress response obtained experimentally.

2.4. Statistical analysis

Four separate statistical analyses, via a linear mixed effectsmodel, are performed using the optimized values for brainstiffness (E1 and E0), relaxation time (τ0), and the degree ofbrain tissue viscoelasticity (α). The linear mixed effects modelis a statistical model that includes both fixed effects (i.e.,postmortem age and hemisphere) and random effects (i.e., pigdemographic). Fixed effects contain factors with levels thatwere of interest for this experiment. All statistical inferencesmade about these factors are confined to the specific levelsstudied. Since the pigs used in the experiments were ran-domly selected from a population of all abattoir aged-pigs,setting pig as a random effect in the model was appropriate.

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Conclusions for the random effect describe the entire popula-tion of levels associated with abattoir-aged pigs. A signifi-cance level of γ¼0.05 was used for all statistical tests. Thebest linear mixed effects model, following an analysis ofvariance (ANOVA) using the software JMP (JMP, 2010),included postmortem age, compressive rate, hemisphere, lobe,binary interaction between postmortem age and compressiverate, and pig number.

3. Results

The fractional Zener model is suitable to describe themechanical behavior of compressed brain tissues subjectedto low strain rates. Fig. 2 shows the FZ fit to the experimentaldata provided by Miller and Chinzei (1997), where unconfinedcompression of brain tissue was performed using a loadingspeed of 5 mm/min (strain rate�0.64�10�2/s). Miller andChinzei (1997) were able to a fit a hyperviscoelastic modelto their experimental dataset. The hyperviscoelastic modelcombines a hyperelastic model (2nd order strain energypotential in polynomial form) and a viscoelastic component(2nd order Prony series in the time domain). For small strains(εo0.15), the FZ model is able to capture the behavior of thebrain tissue dataset provided by Miller and Chinzei (1997). Tocapture the large strain behavior, the fractional Zener modelcan be modified by replacing the branch containing the linearspring (E1) with a nonlinear spring described by a generalizedversion of the Yeoh model. The modified Yeoh model con-siders a fourth-order reduced polynomial hyperelastic model.The strain energy density function (W) for this model isdescribed as follows:

WðI1Þ ¼ ∑n ¼ 4

i ¼ 1CiðI1�3Þi; ð6Þ

where I1 is 1st strain invariant, Ci is material constants.Setting n¼3 (Eq. (6)) reduces the modified Yeoh model to

its standard form, where the order of the reduced polynomialis three (Renaud et al., 2009). The nonlinear spring describedby the fourth-order reduced polynomial Yeoh model provided

0 20 40 60 800

0.4

0.8

1.2

1.6

x 104

Time (s)

Eng

inee

ring

Stre

ss (P

a)

Miller and Chinzei (1997)FZModified FZ (via 4th order Yeoh)

Fig. 2 – Comparison of fit FZ (linear spring), modified FZusing a nonlinear spring described by the 4th order Yeohmodel, and the hyperviscoelastic model by Miller andChinzei (1997).

a better fit to the experimental data described by Miller andChinzei (1997) than its third-order counterpart.

Fig. 2 illustrates the fit of the modified FZ model, using thenonlinear spring described by the fourth-order Yeoh model.By replacing the linear spring with the nonlinear spring, themodified FZ model takes on a similar form to the hypervis-coelastic model described by Miller and Chinzei (1997). Themodified FZ model now contains a hyperelastic component(nonlinear spring) and viscoelastic component (fractionalMaxwell model). The fractional Maxwell branch is describedby the linear spring and fractional element in series. It is thehyperelastic component described by the nonlinear springthat enables the modified FZ model to describe the largestrain behavior of brain tissue subjected to compression.

The hyperviscoelastic model by Miller and Chinzei (1997)was not used to fit our experimental data since only smallstrains were considered. Fig. 2 shows that for small strains (orshort times), the models are similar.

The coefficients (E1, E0, τ0, and α) of the fractional Zenermodel were obtained after application of the nonlinear leastsquares method using the software MATLAB (MATLAB, 2008)to minimize the sum of the square error between thepredicted stress relaxation (Eq. (5)) and the observed stressdue to the ramp-hold strain input generated during theunconfined compression test. Fig. 3 illustrates a typical fit ofthe FZ model to the experimental data.

Replacement of the fractional element (Fig. 1) with adashpot represents the Zener model (standard linear solid),which is not adequate at describing both the ramp and holdphase simultaneously. Fig. 3 demonstrates the inability of theZener model to describe both the loading and relaxationbehavior of the experimental data, as compared with thefractional Zener model. Both the Zener and FZ model are ableto capture the stress response due to the ramp phase.Initially, the stress relaxation due to the hold phase can bedescribed by the Zener model. However, the Zener modelpredicts that relaxation will cease earlier than the experi-mental data dictates. The FZ model overcomes these weak-nesses and can capture the data from the whole experiment(loading and stress relaxation).

0 50 100 1500

40

80

120

160

200

Time (s)

Eng

inee

ring

Stre

ss (P

a)

ExperimentFractional Zener ModelZener Model

Fig. 3 – Typical fit for both the Zener and FZ model whencompared with the experimental data subjected to a ramp-holdstrain input. The FZ coefficients obtained (E1,E0, τ0, and α)that yielded the best fit, will be used in subsequent statisticalanalysis.

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Following parameter optimization, approximately 41% ofthe variablity present in the coefficient E1 was attributed tothe pig used in the experiment. The variability present in theremaining three coefficients was treated as zero due to thenegative variance components that were obtained followingthe Restricted Maximum Likelihood (REML) method. TheREML procedure is used to fit the linear mixed effects model.A different linear mixed effects model may capture thevariability in the dataset using all four coefficients of the FZmodel and yield all positive variance components. Our studyfound that using a different linear mixed effects model didnot yield the “best” linear mixed effects model, for theexperimental dataset.

A statistical analysis was performed, using the materialconstants from the fractional Zener constitutive model (Eq. (5))and the experimental factors, to gain insight into the mechan-ical behavior of degrading swine brain tissue during the ramp-hold strain input. All parameters of the fractional Zener con-stitutive model were approximately normally distributed. Theresults of the linear mixed model are summarized in Table 2.

Table 2 – Summary of results from linear mixed model using ato a strain level of 10%.

Factor Fractional Zener coe

E1

Postmortem age 0.0021Compressive rate 0.6039Hemisphere 0.1675Lobe 0.2571Postmortem agencompressive rate 0.7629

An asterisk (*) implies an interaction between the factors considered.Significant effect of the factor on the fractional Zener model coefficients

Mea

n τ (

s)

0

50

100

150

200

250

0

200

400

600

800

< 6 Hours 24 Hours 3 Days 1 Week

1 mm/min 5 mm/min

< 6 Hours 24 Hours 3 Days 1 Week

1 mm/min 5 mm/min

Mea

n E ∞

(Pa)

1 mm/min 5 mm/min1 mm/min 5 mm/min

Fig. 4 – Mean values of the material constants from the fractioncompressive rate: (a) mean E1; (b) mean E0; (c) mean τ0; and me

Postmortem age and compressive rate significantly affectthe mechanical behavior of the brain tissue. The viscoelasti-city of the brain described by the coefficient α cannot berelated to degradation due to the non-significant effect. Themagnitude of α may be an indicator of the compressive rateused due to the significant effect. The region of sampleextraction did not significantly affect the fractional Zenercoefficients, which agrees with a study by Prange andMargulies (2002) that showed that for small strains, regionaleffects were not significant.

The brain stiffness properties (E1 and E0) are affected bypostmortem age, as evidenced by the significant result in therow containing postmortem age (Table 2). The mean fractionalZener coefficients were used to assess how the materialproperties, such as brain stiffness, and relaxation time (τ0),change by both postmortem age and compressive rate (Fig. 4).For both compressive rates, the magnitude of mean E1decreases as the tissue degrades. This supports the experi-mental observation of softening tissue with increased post-mortem age. A similar trend exists for mean E0, if considering

significance level of γ¼0.05. The dataset used corresponds

fficient

E0 τ0 α

0.0110 0.0772 0.10270.0002 o0.0001 o0.00010.1472 0.1391 0.49230.9229 0.1553 0.79200.0203 0.0161 0.0066

are emphasized using a bold italic font.

Mea

n α

(s)

0

1000

2000

3000

4000

5000

6000

< 6 Hours 24 Hours 3 Days 1 Week

1 mm/min 5 mm/min

< 6 Hours 24 Hours 3 Days 1 Week

1 mm/min 5 mm/min

0.2

0.4

0.6

0.8

1

Mea

n E

0 (P

a)

1 mm/min 5 mm/min

0

al Zener constitutive model by postmortem age andan α. Error bars denote standard deviation.

0

40

80

120

160

200

Time (s)

Eng

inee

ring

Stre

ss (P

a)

0 20 40 60 80 100

< 6 Hours24 Hours3 Days1 Week

0.2

0.4

0.6

0.8

1

Nor

mal

ized

Eng

inee

ring

Stre

ss (u

nitle

ss)

< 6 Hours24 Hours3 Days1 Week

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postmortem age range 24 h–1 week at the 1 mm/min compres-sive rate or o6 h–3 days at the 5 mm/min compressive rate.Tissues experienced an increase in relaxation at lower com-pressive rates, which is why the relaxation time magnitudeswere larger at 1 mm/min than 5 mm/min. There was anincrease in the mean α as the tissue degraded, but it stillranged between 0 and 1. This confirms that the brain tissuestill exhibited viscoelastic behavior during the degradationprocess.

The mean value of material constants were compared tothose optimized by Davis et al. (2006) using compressionexperimental data of human brain tissue (Galford andMcElhaney, 1970) tested within 6–12 h postmortem at lowstrain rates. Table 3 lists these coefficients. The relaxationtime and fractional order were in the same range for tissuestested less than 12 h postmortem.

Fig. 5 shows the FZ model predictions for the four differentpostmortem age cohorts using the mean coefficient valuesprovided by Fig. 4. Brain tissues tested less than 24 h post-mortem are much stiffer than those tested at three or morepostmortem days. This is evidenced by the presence of asteeper slope (ramp region) and higher stress magnitudes(ramp and hold region). A comparison of the FZ modelpredictions for the four different postmortem ages can bemade by first normalizing the stress using the peak stressand normalizing the time using the time associated with thepeak stress (Fig. 5b). During the loading phase, the normal-ized data shows that the postmortem age of the brain tissue isnot discernible. The relaxation behavior attributed to thehold-strain input is the region of interest when characteriza-tion of the brain tissue by postmortem age is needed. Therelaxation behavior between tissue samples that are less than24 h, can overlap initially and may fluctuate due to samplevariability, but will not be softer than samples that are testedgreater than 3 days postmortem. This result agrees with that ofthe literature which showed that the brain is stiffest ifmeasured at least 12 h postmortem (Prange and Margulies,2002).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

Normalized Time (unitless)

Fig. 5 – Mean coefficients from the FZ model were used topredict the material behavior of swine neural tissue for thefour postmortem age groups considered: (a) actual predictionand (b) normalized engineering stress and time data forcomparison of the four postmortem age cohorts.

4. Discussion

Replacement of the dashpot element in the Zener constitu-tive model with a fractional element allowed complex strainhistories (i.e., ramp-hold) to be described using the FZ model.The FZ viscoelastic model was also able to capture the ratedependence characteristic of the brain tissue better than the

Table 3 – Comparison of Fractional Zener coefficients betweentests at low strain rates. All tissues were tested less than 12 h

Reference Test subject Sampl

Bentil (2013) Swine brain tissue (o6 h) �15 m�12 m

Davis et al. (2006) Human brain tissue (6–12 h) �13 m�6 mm

Zener model. These improvements can be quantified whencomparing the norm of the residuals, which is a measure ofthe goodness of the fit between the constitutive modelconsidered and the experimental data (Montgomery, 2009).The norm of the residual was 2.4e3 and 4.73e4 for thefractional Zener and Zener model, respectively. A lower normof the residuals implies that there is a better fit of the modelto the experimental data.

There are similarities and differences between the frac-tional Zener and Zener model from a statistical perspective.The linear mixed model that yielded the most adequateresult for both constitutive models considered comprised of

human and swine brain tissue subjected to compressionpostmortem.

e dimension Model Material constants

m (diameter) Fractional Zener E1¼442 Pam (height) E0¼3520 Pa

τ¼7.62 sα¼0.624

m (diameter) Fractional Zener E1¼1612 Pa(height) E0¼7715 Pa

τ¼6.709 sα¼0.641

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the factors postmortem age, compressive rate, hemisphere,lobe and the binary interaction between postmortem age andcompressive rate. Postmortem age, compressive rate, and theinteraction between postmortem age and compressive ratealso significantly affected the mechanical behavior of thebrain for both the fractional Zener and Zener model. Thestatistical results differ when considering model choice onregional effects. The Zener model showed that samples takenfrom different lobes significantly affected the stress response,but hemisphere did not. Non-significant effects for both thehemisphere and lobe were obtained when the fractionalZener model was used to describe the mechanical behaviorof brain tissue. The FZ model conclusion agreed with theresult by Prange and Margulies (2002), which showed thatregional effects from brain tissue were not significant. Thisstrengthened the argument that the FZ model was animprovement over the Zener model. All statistical resultsfor the Zener model can be found in Bentil (2013).

Although the FZ model is considered to be a linearviscoelastic model since the relaxation form is separable(Davis et al., 2006), it was still able to capture the nonlinearviscoelastic features of the brain tissue at each postmortemtime, making it more robust at describing the behavior ofbrain tissue at low strain rates. The lack of significancebetween the parameter α and the postmortem age, followingan ANOVA, suggests that a correlation does not exist. Statis-tical analysis did confirm that the coefficient α can be used todescribe viscoelastic materials since the parameter α and thecompressive rate factor yielded a statistically significantresult. This agrees with the fact that viscoelastic materialsare sensitive to strain rate and confirms that confining thefractional order to the range 0oαo1 does enables α to betreated as a viscoelastic coefficient.

A comparison of the FZ material constants obtained in thiswork with Davis et al. (2006) showed that the relaxationparameters and the fractional order term were within thesame order of magnitude for tissues tested less than 12 hpostmortem. The coefficients E1 and E0 were less stiff thanthose presented by Davis et al. (2006). Factors that may havecontributed to the discrepancies include the type of animalbrain considered, solution used to store the specimen,dimensions of the specimen, temperature, sample prepara-tion, and test device (Bilston, 2011). A histological workup ofnew tissue samples, to be performed in future research, maygive insight into additional causes of variability between thestiffness parameters with tissue degradation that can becorrelated with the tissue structure. Neurohistology of post-mortem brain tissue have been performed using humansamples taken from patients who suffered from neurodegen-erative diseases, schizophrenia, and even autism(Korschenhausen et al., 1996; Rapp et al., 2006). In suchstudies, normal brain tissue was used as a control to under-stand the abnormal brain, which often differed in the proteincomposition. Currently, the degradation of peptides and brainacidity are some of the parameters that are cited as causingthe changes present in the histology of healthy versusdiseased postmortem brains (Korschenhausen et al., 1996).Studies on the anatomical and physiological changes ofhealthy or even diseased brain changes attributed to post-mortem degradation is lacking and requires additional

research to aid in the ability to link biology with constitutivemodels.

When considering only small strains, the fractional Zenermodel can describe the stress response of brain tissuesubjected to unconfined compression tests performed byother researchers. For instance, a comparison of the frac-tional Zener model with the hyperviscoelastic constitutivemodel developed by Miller and Chinzei (1997) showed thatthe FZ model works as well as, or better than the hypervis-coelastic model, but with fewer material constants to fitresults that consider small strains. If application of thefractional Zener model to brain tissue subjected to largestrains is required, then modification of the branch thatcontains only the linear spring is needed. Replacement ofthis linear spring with a nonlinear spring (i.e. Yeoh 4th ordermodel) enables the modified FZ model to describe braintissue at large strains. This is possible because the modifiedFZ now contains both a hyperelastic and viscoelastic compo-nent, just as the hyperviscoelastic model described by Millerand Chinzei (1997). Although the nonlinear spring in the FZmodel introduces additional constants, the number of con-stants is comparable to the hyperviscoelastic model.

5. Conclusion

The fractional Zener constitutive model is able to capture thenonlinear behavior of brain tissue subjected to unconfinedcompression tests at small strains. The model can be mod-ified such that it can extend to brain tissue subjected to largestrains by replacing the branch containing the linear springwith a nonlinear spring, while keeping the viscoelastic seg-ment unchanged. This study shows the robustness of thefractional Zener model at describing the stress responsechanges of degrading brain tissue by considering only fourmaterial constants. The brain stiffness parameters wereshown to decrease with degradation. The FZ parameter α isintended for describing the viscoelasticity of the brain tissueand should not be used as a metric for quantifying thepostmortem age of the sample.

Acknowledgments

This research was financially supported by the NationalScience Foundation (NSF) under DGE0221678 and CMMI0747252and is gratefully acknowledged.

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