Measuring the anisotropy of the cerebrum in the linear regime

Transcription

1 Measuring the anisotropy of the cerebrum in the linear regime L. Tang MT Coaches: Dr.Ir. J.A.W. van Dommelen Ing. M. Hrapko June 20, 2006

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3 Abstract In this report the anisotropy is measured of the cerebrum in porcine brain tissue. This is done for the three different planes of a brain: transverse, sagittal and coronal. The experiment is done with a dynamic frequency sweep in the linear regime. The samples are measured on an ARES II rotational rheometer in a plate-plate configuration. The samples are placed on the edge of the plate. The advantage of this configuration is that anisotropy can be measured, the measured signal is increased so it allows the measurement of smaller samples and the deformation is more homogenous than in the conventional centered configuration. A sinusoidally strain with a amplitude of 0.01 is applied to the brain tissue in a range of 1 to 10 Hz for all the tests. The transverse plane is almost isotropic. There is a small anisotropy in the sagittal plane and coronal plane. 3

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5 Contents Abstract 3 1 Introduction 7 2 Methods Sample preparation Experiment Post processing Results 13 4 Discussion 21 5 Conclusion 23 List of symbols 25 A Correcting the time dependency 27 References 29 5

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7 Chapter 1 Introduction The human brain stores our memories and determines a great part of our personality. It s also the control center of the whole human body. Therefore the brains are the most important and complex organ of the human body. The brain can be injured by a lot of causes. One of the biggest cause is from motor vehicles accidents. For instance in the United States from 1995 to 2001 an average of 1.4 million cases of traumatic brain injury occurred each year, of which 20 % resulted from motor vehicles accidents [4]. In order to decrease the brain injury from motor vehicle accidents, it is necessary to study the response of the human brain during an impact like in a car accident. Many researchers studied the brain and its material properties. The Finite Element models (figure 1.1), which are 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 the Finite 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 reason many researchers study the mechanical behavior of brain tissue. With the experiments on brain tissue the vehicle safety or devices to decrease the occurrence of brain injuries can 7

8 be improved. The results of these experiments are dependant from many factors, like the post-mortem time. That is the time between death and measurement. A short post-mortem time is better for the measurements because brain tissue will degenerate after death. Another factor is the region of the brain that is used for the experiments. Furthermore animal brains are frequently used as substitute of human brains. All these factors lead to differences in the measured data of various experiments. In this report the directional dependence in the viscous and elastic behavior of the cerebrum of porcine brain tissue is tested. The property of being directionally dependent is called the anisotropy. Something which is anisotropic may have different characteristics, or may appear different in different directions. The anisotropy is tested for the three different planes in a brain: transverse, sagittal and coronal. This is shown in figure 1.2. Figure 1.2: Transverse, sagittal and coronal plane 8

9 Chapter 2 Methods The purpose of the research of brain tissue is to predict the response of the human brain during impact. During the experiments human brain tissue isn t used because the post-mortem time of human brain tissue is too long and the availability is low. Therefore porcine brain tissue is chosen as a substitute for human brain tissue, because the post-mortem time can be minimized and 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 were obtained. 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 a polystyrene box filled with ice. This was done to prevent the dehydration and to slow down the process of degradation of the porcine brain tissue. In the biological laboratory the jar with brains was removed from the box and placed into the Clean Air Cabinet. The samples for the tests of this report were taken from white matter of the cerebrum. This part of the brain was separated from the rest of the brain. Then a smaller part of the cerebrum was cut out and glued to one of the plates of a Leica VT1000S Vibrating-blade microtome. Fine slices were cut by the microtome. The slices were cut at a height of 2 mm, because the recommended gap setting for parallel plates in the rotational rheometer used in this study is between 0.5 and 2.0 mm [1]. This will be explained in the next section. With a cork bore with a diameter of 7 mm 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 the three different planes. Table 2.1: The mean time between death and preparation Time of death - preparation [h] Preparation - testing [h] Total [h] Sagittal Coronal Transverse

10 For the sagittal plane five halves of porcine brains were used and seven samples were made from these brains. Six samples were made from two halves in the coronal plane. In the transverse 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 the human brain. By doing experiments with brain tissue during compression, shear, and tension the material properties can be measured. The results of these experiments vary from study to study, within and across the different modes of testing. This variation is one of the aspects related to the anisotropic and inhomogeneous nature of brain tissue. In this report the experiment is performed on the ARES II rotational rheometer. This machine consists of an upper and lower plate which are parallel to each other. A non-standard eccentric 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 that anisotropy can be measured, the measured signal is increased so it allows the measurement of smaller samples and the deformation is more homogenous than in the conventional centered configuration. 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 the sample at the turntable, the upper plate is lowered down until it will touch the upper surface of the sample (figure 2.1). The normal force should be approximately 0.6 g and the sample may not be compressed with a higher force then 1 g. Then a moist chamber is placed over the plates to prevent dehydration of the sample. The test temperature is set on 37 degrees Celsius because this is the temperature of the human body. The sample is rotated with the turntable every 30 degrees. The angle is chosen randomly during the testing to prevent dependency of the 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 frequency sweeps is to obtain the frequency dependent dynamic modulus G in the linear regime of the material. To measure the direction dependent material properties of brain tissue in the linear regime a dynamic excitation is put on the material. A sinusoidally strain with an amplitude of 0.01, which was determined to be the linear viscoelastic limit by Brands et al 10

11 [2] and Nicolle et al [6] is applied to the brain tissue for a frequency range of 1 to 10 Hz for all the tests. 2.3 Post processing From the measured torque M and angle θ the shear stress τ and shear strain γ can be calculated by [7]: τ = MR 2πR 2 1 ( (R R 1) R2 1 8 ) (2.1) γ = θ R h (2.2) where R is the radius of the plate, R 1 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 equations express the strain and the stress: γ = γ 0 sin(ωt) (2.3) τ = Gγ 0 sin(ωt + δ) (2.4) The behavior of brain tissue is viscoelastic with δ between 0 and π 2. To analyze viscoelastic material the shear stress can be decomposed into two waves of the same frequency but with a phase shift of π 2 and amplitude τ 0 and τ 0 [5] respectively. This is represented mathematically in (2.5) and graphically represented in figure 2.2. τ = τ + τ = τ 0 sin ωt + 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 γ 0 sin(ωt) + G γ 0 cos(ωt) (2.6) For the different orientations the dynamic moduli, G and G, are determined for the following frequencies: Hz, 1.59 Hz, 2.52 Hz, 3.99 Hz, 6.34 Hz and Hz. The elastic behavior is represented by G and the viscous behavior is represented by G. The difference in orientation dependence in elastic and viscous behavior is caused by anisotropy. 11

12 Figure 2.2: Graphical representation of a dynamic excitation and the response versus time 12

13 Chapter 3 Results The goal of the experiments is to determine the anisotropy of the cerebrum in porcine brain tissue. For each sample the storage modulus (G ) and the loss modulus (G ) were measured in the frequency range. For example the results of two samples of the sagittal plane are shown in figure 3.1 and figure 3.2. In figure 3.3 and figure 3.4 the results are shown for two samples of the coronal plane. The results of two samples of the transverse plane are shown in figure 3.5 and figure 3.6. After G G Figure 3.1: Polarplot of G and G versus the angle after time correction 13

14 After G G Figure 3.2: Polarplot of G and G versus the angle after time correction After G G Figure 3.3: Polarplot of G and G versus the angle after time correction 14

15 After G G Figure 3.4: Polarplot of G and G versus the angle after time correction After G G Figure 3.5: Polarplot of G and G versus the angle after time correction 15

16 After G 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 real differences in orientation dependence in elastic behavior (G ) and viscous behavior (G ) caused by anisotropy in the planes, an ellipse is fitted in the polarplot for each sample. An example is shown in figure Figure 3.7: Fitted ellipse in a polarplot 16

17 The size of an ellipse is determined by two constants, A and B. The constant A equals the length of the semimajor axis; the constant B equals the length of the semiminor axis. A semimajor axis is one half of the major axis. Likewise, the semiminor axis is one half of the minor 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 of frequencies can be calculated for the three planes. This is plotted in figure 3.9. Semimajor "A" and Semiminor "B" axis of an Ellipse 10 2 A & B [Pa] 10 1 A G Coronal A G Coronal B G Coronal B G Coronal A G Sagittal A G Sagittal B G Sagittal B G Sagittal A G Transverse A G Transverse B G Transverse B G Transverse Frequency [Hz] Figure 3.9: Semimajor and semiminor axis of an ellipse 17

18 With the values A and B, the eccentricity of an ellipse can be calculated. The eccentricity is a number which express the shape of the ellipse. This can be calculated with formule (3.1). e = 1 B2 A 2 (3.1) The eccentricity is a positive number between 0 and 1, being 0 in the case of a circle. The greater the eccentricity is, the larger the ratio of A to B is, and therefore the more elongated the ellipse is. For each plane the eccentricity of the ellipse is calculated for G and G. The result is shown in figure Eccentricity of the orientation dependent storage and loss moduli 0.65 G Coronal G Coronal G Sagittal G Sagittal G Transverse G Transverse Eccentricity [ ] Frequency [Hz] 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 for G 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 The fraction of A and B for G is bigger than the fraction of A and B for G between the three planes. 18

19 1.3 Fraction of A and B between planes 1.2 Difference between planes [ ] G Coronal/Sagittal G Coronal/Sagittal G Coronal/Transverse G Coronal/Transverse G Transverse/Sagittal G Transverse/Sagittal Frequency [Hz] Figure 3.11: Fraction of A and B between planes 19

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21 Chapter 4 Discussion It s difficult to reproduce the experiments of this report, because they are influenced by several errors. Brain tissue is a very soft material and difficult to handle. There are possibilities to create errors during the preparation of the samples. For instance if the speed of the microtome is too high and so the slices were cut too quick, the material may already have suffered a large strain before testing. Furthermore it is difficult to make a sample that is perfectly flat because brain tissue is very soft. This can cause errors in the measurements because the sample is placed between a parallel plate geometry. The samples are kept in PBS and still cooled after preparation. However due to the unknown microstructure in a sample it is possible that the geometry of the sample changes. Instead of a circular form the sample has an oval form. When the sample is tested for anisotropy and rotated 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 placed exactly 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 is variable. Displacement can also occur during the measurements. There is a possibility that the sample will stick lightly to the upper plate when it is too dry. This results in a small displacement. During the experiment the temperature is 37 degrees Celsius. During the preparation time the samples are cooled. It is possible that the material behavior is different if the sample was heated too short. 21

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23 Chapter 5 Conclusion In this report dynamic frequency sweeps were done to study the direction dependent material behavior of the cerebrum. Therefore porcine brain tissues were prepared in the three different planes of the brain: transverse, sagittal and coronal. The elastic behavior is represented by G and the viscous behavior is represented by G. The transverse plane has the lowest values for 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 plane is almost isotropic. There is a small anisotropy in the sagittal plane and coronal plane. 23

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25 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 [ ] 25

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27 Appendix A Correcting the time dependency A set of measurements can be expressed as a function G depending on the time (t) and other dependencies (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 to be 1. A line can be fitted through the data with G(t)G(θ) = A t+b where A is the approximated slope 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 27

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29 References [1] Advanced Rheometric Expansion System ARES, Rheometric Scientific, Instrument manual [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 the brain 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 injury in the united states: Emergency department visits, hospitalizations and deaths. Center for Disease Control and Prevention, National Center for Injury Prevention and Control, Atlanta(GA). [5] Macosko, C.W. (1994). Rheology Principles, measurements and applications. First editon by Wiley-VHC, inc. [6] Nicolle, S., Lounis, M., ans Willinger, R. (2004). Shear properties of brain tissue over a frequency 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). Passive transverse mechanical properties as a function of temperature of rat skeletal muscle in vitro. Biorheology. 29