- Email: [email protected]
Linear viscoelastic properties of bovine brain tissue in shear
~
Biorheology, Vol!4, No.6, pp. 577-585,1997 CopyrightC 1998 Elotvier Science Ltd Printed in the USA. All rights reserved ()()()l).555X/97$17.00 + .00
Pergamon
PH SOOO&-355X(98)00022-5
LINEAR VISCOELASTIC PROPERTIES OF BOVINE BRAIN TISSUE IN SHEAR LYNNE E. BUSTON, ZIZHEN LIU, AND NHAN PHAN-THIEN Department of Mechanical and Mechatronic Engineering. University of Sydney. New South Wales. AUSTRALIA Reprint requests to: Lynne E. Bilston, Department of Mechanical and Mechatronic Engineering. BuildingJ07, University of Sydney. NSW 2006. AUSTRALIA; Fax: +61-2-9351-7060; e-mail: [email protected]
ABSTRACT We report the results from a series of rheological tests of fresh bovine brain tissue. Using a standard Bohlin VOR shear rheometer, shear relaxation and oscillating strain sweep experiments were performed on disks of brain tissue 30 mm in diameter, with a thickness of 1.5-2 mm, The strain sweep experiment showed that the viscoelastic strain limit is of the order of 0.1 % strain. Shear relaxation data do not indicate the presence of a long-term elastic modulus, indicating fluid-like behavior. A relaxation spectrum was calculated by inverting the experimental data and used to predict oscillatory response, which agreed well with measured data. @ 1998 Elsevier Science Ltd Introduction The rheological properties of the brain have interested researchers for more than 30 years. Much of this interest stems from the role that brain mechanics play in traumatic brain injury and the response of the brain to structural diseases such as hydrocephalus. Several studies have measured the mechanical properties of brain tissues from both cadavers and various animal species. Shuck and Advani (1972) were among the first to subject cadaveric brain tissue to oscillating shear in order to measure the complex modulus. They found that cadaveric brain tissue has a strain yield limit of 0.035 at low frequencies, and 0.013 at high frequencies. Small. but not statistically significant. differences were seen between white and gray matter. Galford and McElhaney (1970) performed tensile creep and tensile stress relaxation tests on human and monkey brain tissues. They found large intersample variations in these properties. and found peak relaxation moduli of the order of 5-9 kPa, with long-term moduli of approximately 1 kPa. Estes and McElhaney (1970) measured the stress-strain responses of human and monkey brain tissues in uniaxial compression and found them to be nonlinear at large strains. More recently. Chinzei and Miller (1996) performed similar compressive stressrelaxation experiments on swine brain tissue, that demonstrated a strong dependence of material response on strain rate, and Arbogast et al. (1995) KEYWORDS:
Biomechanics; rheology; tissue mechanics; viscoelasticity
377
378
Rheological properties of brain tissue
Vol. 34, No.6
characterized swine brain stem using oscillatory pure shear experiments. There is, however, a wide variability in the reported test data. Part of this stems from the differences in experimental technique (relaxation, oscillation, pure shear, compression, different loading rates, strains, etc.) but the nonlinear viscoelastic behavior of brain tissue also plays a role. Establishing a reliable set of brain properties within the linear viscoelastic region is the primary goal of this paper. Brain tissue consists of a complex network of neurons and blood vessels interspersed with a matrix of support cells, all bathed in cerebrospinal fluid. This complex structure and interstitial fluid flow result in nonlinear viscoelastic mechanical properties and large intersample variations compared to engineering materials. Regional variations, particularly between the white and gray matter, have also been reported (Arbogast et al., 1995; Shuck and Advani 1972). Many advances have been made in the mechanical modeling of soft biological tissues. The majority of this work has focused on the behavior of articular cartilage and tendons, although theories developed for these tissues have been extensively used on other tissues. The biphasic and triphasic theories of Mow et at. (1993) use the microscopic structure as the basis, incorporating the elastic collagen matrix and interstitial fluid flow (and, in triphasic theory, electrostatic interactions between long chain molecules in solution). Simon et at. (1993) have used the classic poroelasticity theory in a similar manner, and more recently, allowed for a hyperelastic fiber structure within a liquid medium. Simon (1992) has also demonstrated the equivalence of linear poroelasticity and biphasic theory. These two approaches have been successful in modeling the behavior of cartilage, tendon, myocardium, skin, and intervertebral disks. Recent work by Miller and Chinzei (1995) has shown that the biphasic theory does not appear to be appropriate for the modeling of brain tissue behavior, because it does not allow for the large variation observed in properties with strain rate. The appropriateness of the newer triphasic theory for modeling brain tissue mechanics has not yet been tested. The other common approach to modeling soft tissues is the quasilinear theory developed by Fung (see Fung (1993) for review). This theory requires that the stress response of the material is a separable function of strain and time, and thus that the Boltzmann superposition principle may be used. This method can then be used to model the response of the tissue over a wide range of strain and loading rates by defining both strain and relaxation functions. In the linear viscoelastic region, more traditional rheological descriptions of material response can be used, such as the relaxation spectrum. Methods
Sample preparation Fresh bovine brains were obtained from a commercial abattoir at the time of sacrifice. They were immediately bathed in artificial cerebrospinal fluid (ACSF) solution and placed in a cold container for transport to the testing laboratory. The ACSF was used to prevent swelling due to osmotic effects. Prior to testing. a sample from the deep white matter of the brain was sliced into a 30 mm diameter disk, 1.5 mm thick, using a custom jig. Samples were inspected visually under a dissecting microscope to ensure that they had not been damaged during the cutting process. Sandpaper disks were glued to each parallel plate surface of a Bohlin VOR rheometer to prevent slippage between the sample and the disks during testing. The samples were then irrigated again with ACSF, and placed between the parallel plates of the rheometer. The top plate of the rheometer was lowered to just contact the top of the sample. The
Vol. 34, No.6
Rheological properties of brain tissue
379
outer rim of the sample was coated in petroleum jelly to prevent moisture loss during testing. All samples were tested within 8 h of harvesting to minimize property changes due to microbiological decay. The sample temperature was brought to 37°C and maintained for the duration of the test. The Bohlin VOR rheometer is a controlled strain instrument, with a 12.206 g-em torsion bar, operating in a parallel disk configuration.
Shear relaxation tests Trials were conducted with samples which indicated that a rmmmum of 5-8 cycles of preconditioning were needed to ensure a repeatable shear relaxation result. Thus, prior to shear relaxation tests, samples were preconditioned with 10 cycles of loading at a frequency of 1-2 Hz with the same peak strain to be used in the relaxation test. The sample was then subjected to a sudden (0.02 s rise time) shear strain with peak strain in the range 0.0001-0.07, and held at this strain for 3000 s, while torque data was recorded by the rheometer. The sample was then removed from the rheometer and inspected under a dissecting microscope to determine if any damage occurred during the test.
Strain sweep experiment To determine the linear viscoelastic strain limit for brain tissue, a strain sweep experiment was performed. In this experiment, a sample (prepared as above) was placed in the rheometer, and oscillated at one of three constant frequencies (1,5,20 Hz) while the strain was gradually increased in amplitude. There was a rest period of 10 s between each oscillation. The storage and loss moduli were calculated from the torque data by the rheometer. The strain at which these quantities began to vary with frequency was identified as the linear viscoelastic strain limit. The data from the nonlinear regime has not been used other than to identify the linear viscoelastic strain limit Indeed the response in this nonlinear region mayor may not be sinusoidal, and thus requires more careful analysis to calculate accurate storage and loss moduli.
Experimental Results In each set of data mentioned hereafter, no gross damage was seen in the specimen as a result of the test. Any specimens that showed signs of failure were excluded. Results from relaxation tests at three strain levels are shown in Fig. 1. The data and error bars represent the mean and standard deviation from five data sets. The shape of the relaxation modulus is very similar between different strain levels. although the magnitude varies for strains above 0.1 %. At decreasing peak strains, the data begins to collapse onto the linear viscoelastic response. There is little evidence of a plateau in the relaxation modulus at long times, indicating either that there is no long-term elastic modulus for brain tissue, or that the modulus has a value below 100 Pa, and is thus below the sensitivity of the Bohlin rheometer used here. This will be discussed again later during the analysis. The response was consistent for relaxation times up to approximately 100 s, however at longer times, some variability was seen in the response of different samples. Some samples showed a sharp decrease in the relaxation modulus, while others showed a slight increase. In the first case, water seeping out of the sample may have caused slippage. The latter case may have been due to swelling of the sample. The results from the strain sweep experiment are shown in Fig. 2. Bovine brain appears to have a linear viscoelastic strain limit of approximately 0.1 %, as this is the strain level at which
380
Vol. 34, No.6
Rheological properties Of brain tissue
rocco
-~
(il1000 Q.
0.08% 0.24% .- -6- ..... 0.42% . •. 0-•.•
100
10
0.1
0.01
100
Time (s)
Fig. 1. Stress relaxation data for brain samples at three applied strains. The experimental relaxation modulus. R(t), for samples at peak strains of 0.08%, 0.24%, 0.42% are shown. The data represent the average over five tests, and the error bars (shown only on the data for 0.08%. for clarity) represent one standard deviation for these five tests. the storage and loss moduli begin to vary. This does not appear to be frequency dependent in the 1-20 Hz frequency range tested.
Analysis The relaxation spectrum of a material is often used as a material function in rheology, as it enables a large volume of data to be compressed into a single material function, which is a 'signature' for that material in the linear viscoelastic region. The relaxation spectrum, H('A.), is given by:
--
which is a generalization of the discrete spectrum of the Maxwell model to a continuous function. For our relaxation test, this expression may be written in terms of our relaxation data function R(t) as follows (see Phan-Thien et al. (1997) for derivation): oof
Ci;+ --
R(t)=
00
(£+
JH(A) , --
where:
GE IT
'A. R(t)
H(A)
= =
= =
1- e-(;.
~
dlnA
O~ t~ t;
A..
-tj{
~-l,-XdlnA
t,
i
z»,
A..
long-term elastic modulus rise time of the strain loading dummy variable experimental relaxation modulus.
381
Rheological properties of brain tissue
Vol. 34, No.6
(a) 2000
~
. .a ...~
G'
.r.6~
O···-O··-a:D "
1800
Cbo~ ~~
,~
-;- 1600 D. ......
-0-
1Hz
.... 0-...
5Hz
···6,····20Hz
II)
-S
1400
"
"C
o ~ 1200 1000
800 L-_-'--""""-"""'-'.....................L..-_-'--""""--'--'.....................L..-_.........- -I 1E-4 1E-3 0.01
Strain
(b) 600....-------------------, 500
--......
400
III
D.
II)
::J
300
"5 "C
G"
~ 200
--0-
....0-•..
100
... -t>, ..
1Hz 5Hz 20Hz
Strain Fig. 2. Strain sweep data (a) the storage modulus, G'; (b) the loss modulus, G". The response of brain tissue to a strain sweep experiment, in which the samples are successively loaded to an increasing peak strain at frequencies of I, 5 and 20 Hz is shown. The response is constant for all frequencies until approximately 0.1% strain. This indicates that the linear viscoelastic limit for the tissue is approximately 0.1 % strain.
382
Vol. 34, No.6
Rheological properties of brain tissue
To obtain the relaxation spectrum, H(')..), these integral equations must be inverted. Unfortunately. this is an ill-conditioned numerical problem. since these equations are Fredholm integral equations of the first kind (Weese. 1993) . Small variations in the data can result in large variations in the calculated spectrum. Regularisation methods have been developed to assist in the inversion of these types of equations, which are discussed elsewhere (Weese, 1993). The method of Weese has been employed here to invert the relaxation data to obtain a relaxation spectrum. The effect of noise in the data has been investigated by adding simulated noise to the data and determining the effect on the resulting spectrum. The calculated spectrum is shown in Fig. 3. and the error bars indicate the standard deviation in the spectrum as a result of this simulated noise being added to the data before inversion. Details of this procedure are described in detail by Phan-Thien et al. (1997) . The required long-term modulus, GE, was estimated to be zero, based on the absence of an observable plateau in the relaxation data. Useful descriptive parameters for the relaxation spectrum are the first and second moments of the spectrum: DO
J).H(s)tts
).1
= ~o"--_ _
00
f
).2 H ( s)ds
;.,'/=-,,-0
00
_
DO
f H(s)tts
f H(s)tts
o
o
The mean relaxation time, II, calculated in this manner, for the cattle brains was 79.6 s. The mean square, I 2 , was 85.2 s. To demonstrate the reliability of the data and the robustness of the relaxation spectrum calculated, the original test data was reconstructed from
(U
3
!:, E
So
2
0W C
,g 1 III
)(
III
"iii 0::
01
.Q
0
1E·3
0.01
0.1
1
10
100
1000
Time (5)
Fig. S. Relaxation spectrum calculated from four sets of data taken at a peak strain of 0.1% (the linear viscoelastic limit). The error bars represent the standard deviation in the calculated spectrum due to simulated noise in the data. The vertical dashed lines indicate the region in which the data used lies. Outside these regions, the spectrum may not be reliable.
Vol. 34, No.6
Rheological properties of brain tissue
383
the relaxation spectrum, along with predicted data for cyclic tests in this linear viscoelastic region. These predicted data were calculated by numerically integrating the expression for the experimental relaxation modulus shown above. These were then compared to experimental relaxation data and experimental oscillatory data measured at strains within the linear viscoelastic limit, as shown in Fig. 4(a) and (b). The relaxation data showed good correlation, indicating that the numerical inversion procedure is adequate and robust. There was excellent agreement for the loss modulus, G", and acceptable agreement for the storage modulus, G', although the predicted G' data slightly underestimated the low frequency data. Thus, the spectrum is useful for predicting the response from loading histories other than the shear relaxation from which it was constructed. The latter point means that the spectrum is a good representation of the material response of bovine brain tissue in shear, within the linear viscoelastic range. The sharply decreasing G", with a G' plateau at high frequencies indicates a sharply decreasing loss tangent, and thus a highly elastic response for brain tissue at high frequencies. Discussion As with any set of experimental tests on biological material, there was considerable intersample variability in our data. There are many possible sources of this variability, including the structure of donor animal tissue and donor age as biological variation. Some variability due to the sample preparation is also inevitable, especially small variations in sample thickness within each slice. Due to the heterogeneous and anisotropic nature of the brain, small variations in either the site or angle at which the sample is taken from the brain may cause variations in the mechanical properties, due to orientation of fiber tracts, and small amounts of gray matter appearing in some samples, although this was controlled as much as possible. The data reported shows good consistency between the different test methods (relaxation, oscillation) and acceptable variations in the measured responses (standard deviations of approximately 20% of the average response) for biological samples which are notoriously variable. These factors suggest that the tests are a reasonable representation of the shear response of in vitro brain tissue. The data reported here is similar in many respects to. that reported by other investigators in the literature. The early work of Shuck and Advani (1972) characterized human brain tissue as behaving in a fluid-like manner, which we also found, as indicated by the absence of a long-term elastic modulus. Strain sweep data similar to ours was conducted only up to peak strains of 0.035%, but was linear throughout this range, which is in line with our findings. Shuck and Advani went on to fit a linear viscoelastic model to their data, which fit well up to peak strains of approximately 0.1 %, which, according to our data, is approximately the linear viscoelastic limit. Galford and McElhaney (1970) measured the viscoelastic behavior of both human and monkey brain tissue, and their tensile relaxation data is very similar to our findings (peak relaxation modulus values and decay curves are comparable), although they do not report the strain levels for the relaxation tests. They suggest that at larger strains, 20-40%, the brain response is nonlinearly viscoelastic, which fits in with our findings. This study indicates that the brain exhibits nonlinear viscoelastic behavior from very low strains (0.1%). This is important for modeling the response of brain tissue to mechanical loading, both under injurious loading and in structural brain diseases, strongly suggesting that linear viscoelastic material properties often used in finite element models may be inadequate. Further
384
Vol. 34, No.6
Rheological properties of brain tissue
(a)
10000
Ul
:J
"5
"o
~
c
o
~ )(
• Experimental ••••- Reconstructed
t'CI (jj
a:
.. 100
10
1 Time (5)
(b) 10000.,......--------------------........,
-
1000
t'CI
ll.
...... Ul
:J
"5
" 0
100
. - GO Experimental ....•.... G" Experimental - - GO Calculated ......... G" Calculated
~
10 1E-3
0.01
0.1
10
100
1000
Frequency (Hz)
Fig. 4. Reconstructed shear behavior as predicted by the linear viscoelastic relaxation spectrum. (a) The original shear relaxation data. The error bars are the standard deviation in the experimental data. (b) The calculated response to oscillatory shear, compared with experimental data for oscillatory shear. The line represents the calculated response and the filled symbols represent the experimental data. The error bars on the calculated response represent the uncertainty in the numerical calculation due to simulated noise being introduced into the data before calculating the relaxation spectrum. Experimental error bars in (b) and some points of experimental data are omitted in (a) for clarity.
Vol. 34, No.6
Rheological properties of brain tissue
385
work is needed to characterize the nature of the nonlinear response, as well as the variation with brain region and anatomical direction. It is important that the difference between the mechanical properties of live and post mortem brain tissue also be examined. Tissue perfusion, in particular, is likely to strongly affect the mechanical response of the brain. In vitro testing, such as that carried out in this paper, cannot address this important issue.
References ARBOGAST, K.B., MEANEY, D.F., and THIBAULT, L.E. (1995). Biomechanical characterization of the constitutive relationship for the brainstem. SAE Transactions, Paper 952716, 153-159. CHINZEI, K., and MILLER, K. (1996). Compression of swine brain tissue: experiment in vitro. J Mech. Eng. Lab. 50, 106-115. ESTES, M.S., and MCELHANEY, J.H. (1970). Response of brain tissue to compressive loading. ASME Paper 70-BHF-lJ. FUNG, Y.C. (1993). Biomechanics: Mechanical Properties of Living Tissues. Springer-Verlag, New York, 568 pp. GALFORD, J.E., and MCELHANEY, J.H. (1970) . A viscoelastic study of scalp, brain, and dura. J Biomech. 3, 211-221. MILLER, K., and CHINZEI, K.. (1995). Modeling of soft tissues deformation. J Comput. Aided Surg. 1 (Suppl), 62-63. MOW, V., ATESHIAN, G., and SPILKER, R. (1993). Biomechanics of diarthrodialjoints: a review of twenty years of progress. J Biomech. Eng. 115,460-467. PHAN-THIEN, N., SAFARI-ARDI, M., and MORALES-PATINO, A. (1997) . Oscillatory and simple shear flows of a flour-water dough: a constitutive model. Rheologica Acta 36, 38-48. SHUCK, L.Z., and ADVANI, S.H. (1972). Rheological response of human brain tissue in shear. J Basic Eng. 94, 905-911. SIMON, B.R. (1992). Multiphase poroelastic finite element models for soft tissue structures. Appl. Meek. Rev. 45, 191-218. SIMON, B.R., KAUFMANN, M.V., MCAFEE, M.A., and BALDWIN, A.L. (1993). Determination of material properties for soft tissues using a porohyperelastic constitutive law. Adv. Bioengineering 26, 7-10. WEESE,]. (1993) . A regularization method for nonlinear ill-posed problems. Comput. Physics Communications 77, 429-440. Received 27 August 1997; accepted in revised fonn 10 February 1998.
Biorheology, Vol!4, No.6, pp. 577-585,1997 CopyrightC 1998 Elotvier Science Ltd Printed in the USA. All rights reserved ()()()l).555X/97$17.00 + .00
Pergamon
PH SOOO&-355X(98)00022-5
LINEAR VISCOELASTIC PROPERTIES OF BOVINE BRAIN TISSUE IN SHEAR LYNNE E. BUSTON, ZIZHEN LIU, AND NHAN PHAN-THIEN Department of Mechanical and Mechatronic Engineering. University of Sydney. New South Wales. AUSTRALIA Reprint requests to: Lynne E. Bilston, Department of Mechanical and Mechatronic Engineering. BuildingJ07, University of Sydney. NSW 2006. AUSTRALIA; Fax: +61-2-9351-7060; e-mail: [email protected]
ABSTRACT We report the results from a series of rheological tests of fresh bovine brain tissue. Using a standard Bohlin VOR shear rheometer, shear relaxation and oscillating strain sweep experiments were performed on disks of brain tissue 30 mm in diameter, with a thickness of 1.5-2 mm, The strain sweep experiment showed that the viscoelastic strain limit is of the order of 0.1 % strain. Shear relaxation data do not indicate the presence of a long-term elastic modulus, indicating fluid-like behavior. A relaxation spectrum was calculated by inverting the experimental data and used to predict oscillatory response, which agreed well with measured data. @ 1998 Elsevier Science Ltd Introduction The rheological properties of the brain have interested researchers for more than 30 years. Much of this interest stems from the role that brain mechanics play in traumatic brain injury and the response of the brain to structural diseases such as hydrocephalus. Several studies have measured the mechanical properties of brain tissues from both cadavers and various animal species. Shuck and Advani (1972) were among the first to subject cadaveric brain tissue to oscillating shear in order to measure the complex modulus. They found that cadaveric brain tissue has a strain yield limit of 0.035 at low frequencies, and 0.013 at high frequencies. Small. but not statistically significant. differences were seen between white and gray matter. Galford and McElhaney (1970) performed tensile creep and tensile stress relaxation tests on human and monkey brain tissues. They found large intersample variations in these properties. and found peak relaxation moduli of the order of 5-9 kPa, with long-term moduli of approximately 1 kPa. Estes and McElhaney (1970) measured the stress-strain responses of human and monkey brain tissues in uniaxial compression and found them to be nonlinear at large strains. More recently. Chinzei and Miller (1996) performed similar compressive stressrelaxation experiments on swine brain tissue, that demonstrated a strong dependence of material response on strain rate, and Arbogast et al. (1995) KEYWORDS:
Biomechanics; rheology; tissue mechanics; viscoelasticity
377
378
Rheological properties of brain tissue
Vol. 34, No.6
characterized swine brain stem using oscillatory pure shear experiments. There is, however, a wide variability in the reported test data. Part of this stems from the differences in experimental technique (relaxation, oscillation, pure shear, compression, different loading rates, strains, etc.) but the nonlinear viscoelastic behavior of brain tissue also plays a role. Establishing a reliable set of brain properties within the linear viscoelastic region is the primary goal of this paper. Brain tissue consists of a complex network of neurons and blood vessels interspersed with a matrix of support cells, all bathed in cerebrospinal fluid. This complex structure and interstitial fluid flow result in nonlinear viscoelastic mechanical properties and large intersample variations compared to engineering materials. Regional variations, particularly between the white and gray matter, have also been reported (Arbogast et al., 1995; Shuck and Advani 1972). Many advances have been made in the mechanical modeling of soft biological tissues. The majority of this work has focused on the behavior of articular cartilage and tendons, although theories developed for these tissues have been extensively used on other tissues. The biphasic and triphasic theories of Mow et at. (1993) use the microscopic structure as the basis, incorporating the elastic collagen matrix and interstitial fluid flow (and, in triphasic theory, electrostatic interactions between long chain molecules in solution). Simon et at. (1993) have used the classic poroelasticity theory in a similar manner, and more recently, allowed for a hyperelastic fiber structure within a liquid medium. Simon (1992) has also demonstrated the equivalence of linear poroelasticity and biphasic theory. These two approaches have been successful in modeling the behavior of cartilage, tendon, myocardium, skin, and intervertebral disks. Recent work by Miller and Chinzei (1995) has shown that the biphasic theory does not appear to be appropriate for the modeling of brain tissue behavior, because it does not allow for the large variation observed in properties with strain rate. The appropriateness of the newer triphasic theory for modeling brain tissue mechanics has not yet been tested. The other common approach to modeling soft tissues is the quasilinear theory developed by Fung (see Fung (1993) for review). This theory requires that the stress response of the material is a separable function of strain and time, and thus that the Boltzmann superposition principle may be used. This method can then be used to model the response of the tissue over a wide range of strain and loading rates by defining both strain and relaxation functions. In the linear viscoelastic region, more traditional rheological descriptions of material response can be used, such as the relaxation spectrum. Methods
Sample preparation Fresh bovine brains were obtained from a commercial abattoir at the time of sacrifice. They were immediately bathed in artificial cerebrospinal fluid (ACSF) solution and placed in a cold container for transport to the testing laboratory. The ACSF was used to prevent swelling due to osmotic effects. Prior to testing. a sample from the deep white matter of the brain was sliced into a 30 mm diameter disk, 1.5 mm thick, using a custom jig. Samples were inspected visually under a dissecting microscope to ensure that they had not been damaged during the cutting process. Sandpaper disks were glued to each parallel plate surface of a Bohlin VOR rheometer to prevent slippage between the sample and the disks during testing. The samples were then irrigated again with ACSF, and placed between the parallel plates of the rheometer. The top plate of the rheometer was lowered to just contact the top of the sample. The
Vol. 34, No.6
Rheological properties of brain tissue
379
outer rim of the sample was coated in petroleum jelly to prevent moisture loss during testing. All samples were tested within 8 h of harvesting to minimize property changes due to microbiological decay. The sample temperature was brought to 37°C and maintained for the duration of the test. The Bohlin VOR rheometer is a controlled strain instrument, with a 12.206 g-em torsion bar, operating in a parallel disk configuration.
Shear relaxation tests Trials were conducted with samples which indicated that a rmmmum of 5-8 cycles of preconditioning were needed to ensure a repeatable shear relaxation result. Thus, prior to shear relaxation tests, samples were preconditioned with 10 cycles of loading at a frequency of 1-2 Hz with the same peak strain to be used in the relaxation test. The sample was then subjected to a sudden (0.02 s rise time) shear strain with peak strain in the range 0.0001-0.07, and held at this strain for 3000 s, while torque data was recorded by the rheometer. The sample was then removed from the rheometer and inspected under a dissecting microscope to determine if any damage occurred during the test.
Strain sweep experiment To determine the linear viscoelastic strain limit for brain tissue, a strain sweep experiment was performed. In this experiment, a sample (prepared as above) was placed in the rheometer, and oscillated at one of three constant frequencies (1,5,20 Hz) while the strain was gradually increased in amplitude. There was a rest period of 10 s between each oscillation. The storage and loss moduli were calculated from the torque data by the rheometer. The strain at which these quantities began to vary with frequency was identified as the linear viscoelastic strain limit. The data from the nonlinear regime has not been used other than to identify the linear viscoelastic strain limit Indeed the response in this nonlinear region mayor may not be sinusoidal, and thus requires more careful analysis to calculate accurate storage and loss moduli.
Experimental Results In each set of data mentioned hereafter, no gross damage was seen in the specimen as a result of the test. Any specimens that showed signs of failure were excluded. Results from relaxation tests at three strain levels are shown in Fig. 1. The data and error bars represent the mean and standard deviation from five data sets. The shape of the relaxation modulus is very similar between different strain levels. although the magnitude varies for strains above 0.1 %. At decreasing peak strains, the data begins to collapse onto the linear viscoelastic response. There is little evidence of a plateau in the relaxation modulus at long times, indicating either that there is no long-term elastic modulus for brain tissue, or that the modulus has a value below 100 Pa, and is thus below the sensitivity of the Bohlin rheometer used here. This will be discussed again later during the analysis. The response was consistent for relaxation times up to approximately 100 s, however at longer times, some variability was seen in the response of different samples. Some samples showed a sharp decrease in the relaxation modulus, while others showed a slight increase. In the first case, water seeping out of the sample may have caused slippage. The latter case may have been due to swelling of the sample. The results from the strain sweep experiment are shown in Fig. 2. Bovine brain appears to have a linear viscoelastic strain limit of approximately 0.1 %, as this is the strain level at which
380
Vol. 34, No.6
Rheological properties Of brain tissue
rocco
-~
(il1000 Q.
0.08% 0.24% .- -6- ..... 0.42% . •. 0-•.•
100
10
0.1
0.01
100
Time (s)
Fig. 1. Stress relaxation data for brain samples at three applied strains. The experimental relaxation modulus. R(t), for samples at peak strains of 0.08%, 0.24%, 0.42% are shown. The data represent the average over five tests, and the error bars (shown only on the data for 0.08%. for clarity) represent one standard deviation for these five tests. the storage and loss moduli begin to vary. This does not appear to be frequency dependent in the 1-20 Hz frequency range tested.
Analysis The relaxation spectrum of a material is often used as a material function in rheology, as it enables a large volume of data to be compressed into a single material function, which is a 'signature' for that material in the linear viscoelastic region. The relaxation spectrum, H('A.), is given by:
--
which is a generalization of the discrete spectrum of the Maxwell model to a continuous function. For our relaxation test, this expression may be written in terms of our relaxation data function R(t) as follows (see Phan-Thien et al. (1997) for derivation): oof
Ci;+ --
R(t)=
00
(£+
JH(A) , --
where:
GE IT
'A. R(t)
H(A)
= =
= =
1- e-(;.
~
dlnA
O~ t~ t;
A..
-tj{
~-l,-XdlnA
t,
i
z»,
A..
long-term elastic modulus rise time of the strain loading dummy variable experimental relaxation modulus.
381
Rheological properties of brain tissue
Vol. 34, No.6
(a) 2000
~
. .a ...~
G'
.r.6~
O···-O··-a:D
1800
Cbo~ ~~
,~
-;- 1600 D. ......
-0-
1Hz
.... 0-...
5Hz
···6,····20Hz
II)
-S
1400
"
"C
o ~ 1200 1000
800 L-_-'--""""-"""'-'.....................L..-_-'--""""--'--'.....................L..-_.........- -I 1E-4 1E-3 0.01
Strain
(b) 600....-------------------, 500
--......
400
III
D.
II)
::J
300
"5 "C
G"
~ 200
--0-
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Strain Fig. 2. Strain sweep data (a) the storage modulus, G'; (b) the loss modulus, G". The response of brain tissue to a strain sweep experiment, in which the samples are successively loaded to an increasing peak strain at frequencies of I, 5 and 20 Hz is shown. The response is constant for all frequencies until approximately 0.1% strain. This indicates that the linear viscoelastic limit for the tissue is approximately 0.1 % strain.
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To obtain the relaxation spectrum, H(')..), these integral equations must be inverted. Unfortunately. this is an ill-conditioned numerical problem. since these equations are Fredholm integral equations of the first kind (Weese. 1993) . Small variations in the data can result in large variations in the calculated spectrum. Regularisation methods have been developed to assist in the inversion of these types of equations, which are discussed elsewhere (Weese, 1993). The method of Weese has been employed here to invert the relaxation data to obtain a relaxation spectrum. The effect of noise in the data has been investigated by adding simulated noise to the data and determining the effect on the resulting spectrum. The calculated spectrum is shown in Fig. 3. and the error bars indicate the standard deviation in the spectrum as a result of this simulated noise being added to the data before inversion. Details of this procedure are described in detail by Phan-Thien et al. (1997) . The required long-term modulus, GE, was estimated to be zero, based on the absence of an observable plateau in the relaxation data. Useful descriptive parameters for the relaxation spectrum are the first and second moments of the spectrum: DO
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The mean relaxation time, II, calculated in this manner, for the cattle brains was 79.6 s. The mean square, I 2 , was 85.2 s. To demonstrate the reliability of the data and the robustness of the relaxation spectrum calculated, the original test data was reconstructed from
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Fig. S. Relaxation spectrum calculated from four sets of data taken at a peak strain of 0.1% (the linear viscoelastic limit). The error bars represent the standard deviation in the calculated spectrum due to simulated noise in the data. The vertical dashed lines indicate the region in which the data used lies. Outside these regions, the spectrum may not be reliable.
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the relaxation spectrum, along with predicted data for cyclic tests in this linear viscoelastic region. These predicted data were calculated by numerically integrating the expression for the experimental relaxation modulus shown above. These were then compared to experimental relaxation data and experimental oscillatory data measured at strains within the linear viscoelastic limit, as shown in Fig. 4(a) and (b). The relaxation data showed good correlation, indicating that the numerical inversion procedure is adequate and robust. There was excellent agreement for the loss modulus, G", and acceptable agreement for the storage modulus, G', although the predicted G' data slightly underestimated the low frequency data. Thus, the spectrum is useful for predicting the response from loading histories other than the shear relaxation from which it was constructed. The latter point means that the spectrum is a good representation of the material response of bovine brain tissue in shear, within the linear viscoelastic range. The sharply decreasing G", with a G' plateau at high frequencies indicates a sharply decreasing loss tangent, and thus a highly elastic response for brain tissue at high frequencies. Discussion As with any set of experimental tests on biological material, there was considerable intersample variability in our data. There are many possible sources of this variability, including the structure of donor animal tissue and donor age as biological variation. Some variability due to the sample preparation is also inevitable, especially small variations in sample thickness within each slice. Due to the heterogeneous and anisotropic nature of the brain, small variations in either the site or angle at which the sample is taken from the brain may cause variations in the mechanical properties, due to orientation of fiber tracts, and small amounts of gray matter appearing in some samples, although this was controlled as much as possible. The data reported shows good consistency between the different test methods (relaxation, oscillation) and acceptable variations in the measured responses (standard deviations of approximately 20% of the average response) for biological samples which are notoriously variable. These factors suggest that the tests are a reasonable representation of the shear response of in vitro brain tissue. The data reported here is similar in many respects to. that reported by other investigators in the literature. The early work of Shuck and Advani (1972) characterized human brain tissue as behaving in a fluid-like manner, which we also found, as indicated by the absence of a long-term elastic modulus. Strain sweep data similar to ours was conducted only up to peak strains of 0.035%, but was linear throughout this range, which is in line with our findings. Shuck and Advani went on to fit a linear viscoelastic model to their data, which fit well up to peak strains of approximately 0.1 %, which, according to our data, is approximately the linear viscoelastic limit. Galford and McElhaney (1970) measured the viscoelastic behavior of both human and monkey brain tissue, and their tensile relaxation data is very similar to our findings (peak relaxation modulus values and decay curves are comparable), although they do not report the strain levels for the relaxation tests. They suggest that at larger strains, 20-40%, the brain response is nonlinearly viscoelastic, which fits in with our findings. This study indicates that the brain exhibits nonlinear viscoelastic behavior from very low strains (0.1%). This is important for modeling the response of brain tissue to mechanical loading, both under injurious loading and in structural brain diseases, strongly suggesting that linear viscoelastic material properties often used in finite element models may be inadequate. Further
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Rheological properties of brain tissue
(a)
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:J
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o
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Fig. 4. Reconstructed shear behavior as predicted by the linear viscoelastic relaxation spectrum. (a) The original shear relaxation data. The error bars are the standard deviation in the experimental data. (b) The calculated response to oscillatory shear, compared with experimental data for oscillatory shear. The line represents the calculated response and the filled symbols represent the experimental data. The error bars on the calculated response represent the uncertainty in the numerical calculation due to simulated noise being introduced into the data before calculating the relaxation spectrum. Experimental error bars in (b) and some points of experimental data are omitted in (a) for clarity.
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work is needed to characterize the nature of the nonlinear response, as well as the variation with brain region and anatomical direction. It is important that the difference between the mechanical properties of live and post mortem brain tissue also be examined. Tissue perfusion, in particular, is likely to strongly affect the mechanical response of the brain. In vitro testing, such as that carried out in this paper, cannot address this important issue.
References ARBOGAST, K.B., MEANEY, D.F., and THIBAULT, L.E. (1995). Biomechanical characterization of the constitutive relationship for the brainstem. SAE Transactions, Paper 952716, 153-159. CHINZEI, K., and MILLER, K. (1996). Compression of swine brain tissue: experiment in vitro. J Mech. Eng. Lab. 50, 106-115. ESTES, M.S., and MCELHANEY, J.H. (1970). Response of brain tissue to compressive loading. ASME Paper 70-BHF-lJ. FUNG, Y.C. (1993). Biomechanics: Mechanical Properties of Living Tissues. Springer-Verlag, New York, 568 pp. GALFORD, J.E., and MCELHANEY, J.H. (1970) . A viscoelastic study of scalp, brain, and dura. J Biomech. 3, 211-221. MILLER, K., and CHINZEI, K.. (1995). Modeling of soft tissues deformation. J Comput. Aided Surg. 1 (Suppl), 62-63. MOW, V., ATESHIAN, G., and SPILKER, R. (1993). Biomechanics of diarthrodialjoints: a review of twenty years of progress. J Biomech. Eng. 115,460-467. PHAN-THIEN, N., SAFARI-ARDI, M., and MORALES-PATINO, A. (1997) . Oscillatory and simple shear flows of a flour-water dough: a constitutive model. Rheologica Acta 36, 38-48. SHUCK, L.Z., and ADVANI, S.H. (1972). Rheological response of human brain tissue in shear. J Basic Eng. 94, 905-911. SIMON, B.R. (1992). Multiphase poroelastic finite element models for soft tissue structures. Appl. Meek. Rev. 45, 191-218. SIMON, B.R., KAUFMANN, M.V., MCAFEE, M.A., and BALDWIN, A.L. (1993). Determination of material properties for soft tissues using a porohyperelastic constitutive law. Adv. Bioengineering 26, 7-10. WEESE,]. (1993) . A regularization method for nonlinear ill-posed problems. Comput. Physics Communications 77, 429-440. Received 27 August 1997; accepted in revised fonn 10 February 1998.