measuring the anisotropy of the cerebrum in the linear regime · the goal of the experiments is to...
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Measuring the anisotropy of thecerebrum in the linear regime
L. TangMT 06.26
Coaches:
Dr.Ir. J.A.W. van DommelenIng. M. Hrapko
June 20, 2006
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Abstract
In this report the anisotropy is measured of the cerebrum in porcine brain tissue. This is donefor the three different planes of a brain: transverse, sagittal and coronal. The experiment isdone with a dynamic frequency sweep in the linear regime. The samples are measured on anARES II rotational rheometer in a plate-plate configuration. The samples are placed on theedge of the plate. The advantage of this configuration is that anisotropy can be measured, themeasured signal is increased so it allows the measurement of smaller samples and the defor-mation is more homogenous than in the conventional centered configuration. A sinusoidallystrain with a amplitude of 0.01 is applied to the brain tissue in a range of 1 to 10 Hz for allthe tests. The transverse plane is almost isotropic. There is a small anisotropy in the sagittalplane and coronal plane.
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Contents
Abstract 3
1 Introduction 7
2 Methods 9
2.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Post processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Results 13
4 Discussion 21
5 Conclusion 23
List of symbols 25
A Correcting the time dependency 27
References 29
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Chapter 1
Introduction
The human brain stores our memories and determines a great part of our personality. It’s alsothe control center of the whole human body. Therefore the brains are the most important andcomplex organ of the human body. The brain can be injured by a lot of causes. One of thebiggest cause is from motor vehicles accidents. For instance in the United States from 1995to 2001 an average of 1.4 million cases of traumatic brain injury occurred each year, of which20 % resulted from motor vehicles accidents [4].
In order to decrease the brain injury from motor vehicle accidents, it is necessary to studythe response of the human brain during an impact like in a car accident. Many researchersstudied the brain and its material properties. The Finite Element models (figure 1.1), whichare being developed nowadays, are used to predict the injury of the brain during an impact.The results of various experiments can be used to improve the predictive capabilities of theFinite Element models.
Figure 1.1: Finite Element model of the brain
In these FE models the mechanical behavior of the human brain is required. For this reasonmany researchers study the mechanical behavior of brain tissue. With the experiments onbrain tissue the vehicle safety or devices to decrease the occurrence of brain injuries can
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be improved. The results of these experiments are dependant from many factors, like thepost-mortem time. That is the time between death and measurement. A short post-mortemtime is better for the measurements because brain tissue will degenerate after death. Anotherfactor is the region of the brain that is used for the experiments. Furthermore animal brainsare frequently used as substitute of human brains. All these factors lead to differences inthe measured data of various experiments. In this report the directional dependence in theviscous and elastic behavior of the cerebrum of porcine brain tissue is tested. The propertyof being directionally dependent is called the anisotropy. Something which is anisotropic mayhave different characteristics, or may appear different in different directions. The anisotropyis tested for the three different planes in a brain: transverse, sagittal and coronal. This isshown in figure 1.2.
Figure 1.2: Transverse, sagittal and coronal plane
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Chapter 2
Methods
The purpose of the research of brain tissue is to predict the response of the human brain duringimpact. During the experiments human brain tissue isn’t used because the post-mortem timeof human brain tissue is too long and the availability is low. Therefore porcine brain tissue ischosen as a substitute for human brain tissue, because the post-mortem time can be minimizedand is easily available [3]. Before the testing, brain samples have to be prepared.
2.1 Sample preparation
From a local slaughterhouse fresh halves of porcine brains from 6-12 month old pigs wereobtained. To transport the halves of porcine brains from the slaughterhouse to the university,the brains were put into a jar with Phosphate Buffered Saline solution (PBS) and placed in apolystyrene box filled with ice. This was done to prevent the dehydration and to slow downthe process of degradation of the porcine brain tissue. In the biological laboratory the jar withbrains was removed from the box and placed into the Clean Air Cabinet. The samples for thetests of this report were taken from white matter of the cerebrum. This part of the brain wasseparated from the rest of the brain. Then a smaller part of the cerebrum was cut out andglued to one of the plates of a Leica VT1000S Vibrating-blade microtome. Fine slices werecut by the microtome. The slices were cut at a height of 2 mm, because the recommended gapsetting for parallel plates in the rotational rheometer used in this study is between 0.5 and2.0 mm [1]. This will be explained in the next section. With a cork bore with a diameter of 7mm circular samples were made out of the slices. The samples were made in the transverse,sagittal and coronal plane. In table 2.1 the post-mortem time of the samples is given for thethree different planes.
Table 2.1: The mean time between death and preparation
Time of death - preparation [h] Preparation - testing [h] Total [h]Sagittal 3.87 1.45 5.32Coronal 5.68 2.95 8.63Transverse 4.38 2.37 6.75
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For the sagittal plane five halves of porcine brains were used and seven samples were madefrom these brains. Six samples were made from two halves in the coronal plane. In the trans-verse plane also six samples were made but from three halves.
2.2 Experiment
In the past the material properties of brain tissue were unknown by the researchers of thehuman brain. By doing experiments with brain tissue during compression, shear, and tensionthe material properties can be measured. The results of these experiments vary from studyto study, within and across the different modes of testing. This variation is one of the aspectsrelated to the anisotropic and inhomogeneous nature of brain tissue.
In this report the experiment is performed on the ARES II rotational rheometer. Thismachine consists of an upper and lower plate which are parallel to each other. A non-standardeccentric configuration was used, where the sample was placed at the edge of the plate [7].This is schematically represented in figure 2.1. The advantage of this configuration is thatanisotropy can be measured, the measured signal is increased so it allows the measurement ofsmaller samples and the deformation is more homogenous than in the conventional centeredconfiguration.
Figure 2.1: Upper and lower plate of the ARES II Rheometer
Sandpaper is glued on these plates to prevent slip during the measurements. After placing thesample at the turntable, the upper plate is lowered down until it will touch the upper surfaceof the sample (figure 2.1). The normal force should be approximately 0.6 g and the samplemay not be compressed with a higher force then 1 g. Then a moist chamber is placed over theplates to prevent dehydration of the sample. The test temperature is set on 37 degrees Celsiusbecause this is the temperature of the human body. The sample is rotated with the turntableevery 30 degrees. The angle is chosen randomly during the testing to prevent dependency ofthe time. There is still some dependency of the time which is corrected (Appendix A).
The experiment is done with a dynamic frequency sweep (DFS). The goal of dynamic fre-quency sweeps is to obtain the frequency dependent dynamic modulus G in the linear regimeof the material. To measure the direction dependent material properties of brain tissue inthe linear regime a dynamic excitation is put on the material. A sinusoidally strain with anamplitude of 0.01, which was determined to be the linear viscoelastic limit by Brands et al
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[2] and Nicolle et al [6] is applied to the brain tissue for a frequency range of 1 to 10 Hz forall the tests.
2.3 Post processing
From the measured torque M and angle θ the shear stress τ and shear strain γ can becalculated by [7]:
τ =MR
2πR21(
(R−R1)2
2 + R21
8 )(2.1)
γ = θR
h(2.2)
where R is the radius of the plate, R1 is the radius of the sample and h is height of the sample.
A sinusoidally strain is applied to the brain tissue for all the test as described in paragraph 2.2.The strain is sinusoidal, so the stress will also respond sinusoidally. The following equationsexpress the strain and the stress:
γ = γ0 sin(ωt) (2.3)
τ = Gγ0 sin(ωt + δ) (2.4)
The behavior of brain tissue is viscoelastic with δ between 0 and π2 . To analyze viscoelastic
material the shear stress can be decomposed into two waves of the same frequency but with aphase shift of π
2 and amplitude τ ′0 and τ ′′0 [5] respectively. This is represented mathematicallyin (2.5) and graphically represented in figure 2.2.
τ = τ ′ + τ ′′ = τ ′0 sinωt + τ ′′0 cos ωt (2.5)
Equation (2.5) can now be re-arranged to (2.6). This leads to the two moduli:G′ (storage modulus) and G′′ (loss modulus). Both moduli are frequency dependent.
τ = G′γ0 sin(ωt) + G′′γ0 cos(ωt) (2.6)
For the different orientations the dynamic moduli, G′ and G′′, are determined for the followingfrequencies: 1.004 Hz, 1.59 Hz, 2.52 Hz, 3.99 Hz, 6.34 Hz and 10.04 Hz. The elastic behavior isrepresented by G′ and the viscous behavior is represented by G′′. The difference in orientationdependence in elastic and viscous behavior is caused by anisotropy.
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Figure 2.2: Graphical representation of a dynamic excitation and the response versus time
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Chapter 3
Results
The goal of the experiments is to determine the anisotropy of the cerebrum in porcine braintissue. For each sample the storage modulus (G′) and the loss modulus (G′′) were measuredin the frequency range. For example the results of two samples of the sagittal plane are shownin figure 3.1 and figure 3.2. In figure 3.3 and figure 3.4 the results are shown for two samplesof the coronal plane. The results of two samples of the transverse plane are shown in figure 3.5and figure 3.6.
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180 0
AfterG’G’’
Figure 3.1: Polarplot of G′ and G′′ versus the angle after time correction
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270
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180 0
AfterG’G’’
Figure 3.2: Polarplot of G′ and G′′ versus the angle after time correction
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AfterG’G’’
Figure 3.3: Polarplot of G′ and G′′ versus the angle after time correction
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AfterG’G’’
Figure 3.4: Polarplot of G′ and G′′ versus the angle after time correction
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AfterG’G’’
Figure 3.5: Polarplot of G′ and G′′ versus the angle after time correction
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AfterG’G’’
Figure 3.6: Polarplot of G′ and G′′ versus the angle after time correction
The differences in the three different planes for G′ and G′′ isn’t really visible. To see realdifferences in orientation dependence in elastic behavior (G′) and viscous behavior (G′′) causedby anisotropy in the planes, an ellipse is fitted in the polarplot for each sample. An exampleis shown in figure 3.7.
−500 −400 −300 −200 −100 0 100 200 300 400
−300
−200
−100
0
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Figure 3.7: Fitted ellipse in a polarplot
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The size of an ellipse is determined by two constants, A and B. The constant A equals thelength of the semimajor axis; the constant B equals the length of the semiminor axis. Asemimajor axis is one half of the major axis. Likewise, the semiminor axis is one half of theminor axis. This is shown in figure 3.8.
Figure 3.8: The constants A and B of an ellipse
Each sample has a value A and B for G′ and G′′. A mean value of A and B for the range offrequencies can be calculated for the three planes. This is plotted in figure 3.9.
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Semimajor "A" and Semiminor "B" axis of an Ellipse
Frequency [Hz]
A &
B [P
a]
A − G’ CoronalA − G’’ CoronalB − G’ CoronalB − G’’ CoronalA − G’ SagittalA − G’’ SagittalB − G’ SagittalB − G’’ SagittalA − G’ TransverseA − G’’ TransverseB − G’ TransverseB − G’’ Transverse
Figure 3.9: Semimajor and semiminor axis of an ellipse
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With the values A and B, the eccentricity of an ellipse can be calculated. The eccentricity isa number which express the shape of the ellipse. This can be calculated with formule (3.1).
e =
√1− B2
A2(3.1)
The eccentricity is a positive number between 0 and 1, being 0 in the case of a circle. Thegreater the eccentricity is, the larger the ratio of A to B is, and therefore the more elongatedthe ellipse is. For each plane the eccentricity of the ellipse is calculated for G′ and G′′. Theresult is shown in figure 3.10.
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0.4
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0.6
0.65
Eccentricity of the orientation dependent storage and loss moduli
Frequency [Hz]
Ecc
entr
icity
[−]
G’ CoronalG’’ CoronalG’ SagittalG’’ SagittalG’ TransverseG’’ Transverse
Figure 3.10: Mean eccentricity of the orientation dependent storage and loss moduli
In the transverse plane the anisotropy has the lowest value for G′ and G′′. The anisotropy forG′ and G′′ in the sagittal and coronal plane is almost the same.
A comparison of the fraction of A and B between the three different planes can also be made.This is shown in figure 3.11. The fraction of A and B for G′ is bigger than the fraction of Aand B for G′′ between the three planes.
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0.8
0.9
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1.1
1.2
1.3Fraction of A and B between planes
Frequency [Hz]
Diff
eren
ce b
etw
een
plan
es [−
]
G’ Coronal/SagittalG’’ Coronal/SagittalG’ Coronal/TransverseG’’ Coronal/TransverseG’ Transverse/SagittalG’’ Transverse/Sagittal
Figure 3.11: Fraction of A and B between planes
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Chapter 4
Discussion
It’s difficult to reproduce the experiments of this report, because they are influenced by severalerrors. Brain tissue is a very soft material and difficult to handle. There are possibilities tocreate errors during the preparation of the samples. For instance if the speed of the microtomeis too high and so the slices were cut too quick, the material may already have suffered a largestrain before testing. Furthermore it is difficult to make a sample that is perfectly flat becausebrain tissue is very soft. This can cause errors in the measurements because the sample isplaced between a parallel plate geometry.
The samples are kept in PBS and still cooled after preparation. However due to the unknownmicrostructure in a sample it is possible that the geometry of the sample changes. Instead ofa circular form the sample has an oval form. When the sample is tested for anisotropy androtated into other directions, the sample is not sheared the same in every direction.
There are also errors caused by the placing of samples. The sample is not always placedexactly in the middle of the turntable of the ARES because the sample is placed by hand.When the turntable is rotated into other directions the distance to the center of the plate isvariable. Displacement can also occur during the measurements. There is a possibility thatthe sample will stick lightly to the upper plate when it is too dry. This results in a smalldisplacement.
During the experiment the temperature is 37 degrees Celsius. During the preparation timethe samples are cooled. It is possible that the material behavior is different if the sample washeated too short.
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Chapter 5
Conclusion
In this report dynamic frequency sweeps were done to study the direction dependent materialbehavior of the cerebrum. Therefore porcine brain tissues were prepared in the three differentplanes of the brain: transverse, sagittal and coronal. The elastic behavior is represented byG′ and the viscous behavior is represented by G′′. The transverse plane has the lowest valuesfor G′ and G′′. In the sagittal plane the value for G′ is higher then in in the coronal plane.The value for G′′ is higher in the coronal plane. The conclusion is that the transverse planeis almost isotropic. There is a small anisotropy in the sagittal plane and coronal plane.
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List of symbols
G Relaxation modulus [Pa]G′ Storage modulus [Pa]G′′ Loss modulus [Pa]M Torque [Nm]R Radius [m]t Time [s]ω Angular [rad/s]γ Shear strain [-]τ Shear tress [Pa]δ Phase shift [rad]θ Angle [◦]
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Appendix A
Correcting the time dependency
A set of measurements can be expressed as a function G depending on the time (t) and otherdependencies (θ): G(t, θ) = G(t)G(θ)Where the function for the time dependency G(t) can be written as:G(t) = At + B where A is the slope and B is the starting point of the line and is chosen tobe 1.A line can be fitted through the data with G(t)G(θ) = A∗t+B∗ where A∗ is the approximatedslope and B∗ is the value where the function begins.The correction can be calculated with (At + 1)G = A∗t + B∗
Thus:A = A∗
G , G = B∗
A = A∗
B∗
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References
[1] Advanced Rheometric Expansion System ARES, Rheometric Scientific, Instrument man-ual
[2] Brands, D.W.A., Bovendeerd, P.H.M., Peters, G.W.M., Wismans, J.S.H.M., Paas,M.H.J.W., and van Bree, J.L.M.J. (1999). Comparison of the dynamic behavior of thebrain tissue and two model materials. Proc. of the 44th Stapp Car Crash Conference,99SC21:57-64
[3] Hrapko, M., Dommelen van, J.A.W., Peters, G.W.M., and Wismans, J.S.H.M. (2005).The mechanical behaviour of brain tissue: large strain response and constitutive modelling.Proceedings of the International Ircobi Conference, Prague, Czech Republic.
[4] Langlois, J.A., Rutland-Brown, W., and Thomas, K.E. (2004). Traumatic brain injuryin the united states: Emergency department visits, hospitalizations and deaths. Centerfor Disease Control and Prevention, National Center for Injury Prevention and Control,Atlanta(GA).
[5] Macosko, C.W. (1994). Rheology Principles, measurements and applications. First editonby Wiley-VHC, inc.
[6] Nicolle, S., Lounis, M., ans Willinger, R. (2004). Shear properties of brain tissue over afrequency range relevant for automotive impact situations: New experimental results. Proc.of the 48th Stapp Car Crash Journal, 48:1-20.
[7] Turnhout van, M., Oomens, C., Peters, G., and Stekelenburg, A. (accepted 2005). Passivetransverse mechanical properties as a function of temperature of rat skeletal muscle in vitro.Biorheology.
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