Modeling of Dynamic Rigidity Modulus of Brain Matter

Transcription

1 The African Review of Physics (016) 11: Modeling of Dynamic Rigidity Modulus of Brain Matter Moez Chakroun * and Med Hédi Ben Ghozlen Materials Physics Laboratory, Faculty of Sciences, Sfax University, Sfax, Tunisia Agar gel is a linear viscoelastic material at low deformations (> 0.01%). The characterization technique used is the rheometer type carri-med "CSL 100". This rheometer offers us the opportunity to study the gel under static and dynamic shear. The study is done at low frequencies (0.1-40Hz). Mechanical characterization of the agar gel in terms of dynamic modulus was realized for different concentrations. The dynamic rigidity of the gel decreases with increasing concentration. The 8% agar gel simulates very well the dynamic rigidity of brain tissue in low frequency. The range of low frequencies is rarely studied for this material (brain) in the literature. Most tests done on brain tissue are in a frequency range between 50 and 600 Hz. The model of Maxwell-Kelvin-Voigt simulates very well the 8% agar gel. The instant elasticity derived from mathematical modeling of agar gel is similar to that measured in the literature to the brain tissue. The agar gel can be used in the construction of physical models of the human head used to analyze the dynamic response of the head to shock or to an inertial load. 1. Introduction The biological tissues are generally considered as soft tissues. These tissues have intermediate properties between perfect elastic solid and Newtonian fluid. These tissues are considered as viscoelastic [1]. Several approaches are proposed in the literature to analyze the dynamic response of the head to shock or to an inertial load. These approaches use a variety of techniques such as human experimentation, physical and mathematical modeling [,3,4,5,6,7] and tests on human [8] or animal [9] anatomical sample. The construction of physical models requires the use of adequate material to simulate various constituents of head tissues. In 1999, Brand [10] tried to simulate the brain matter by gelatin and then by the dielectric silicone gel. This work has not led to good results. Although the silicone gel has a dynamic modulus similar to the brain tissue in the order of magnitude, gelatin has viscoelastic properties very superior to that tissue. Our goal is to find the best concentration of agar gel (polymer) to simulate the mechanical behavior of the brain matter at low frequencies [0-40Hz]. This gel was discovered by Minora Tarazaemon [11]. It is a galactose polymer contained in the cell wall of certain species of red algae (Rhodophyceae). The mechanical properties of agar gel are determined in shear.. Materials and Methods The mechanical tests have been realized using a Carri-Med rheometer "CSL 100". This rheometer consists principally of two parallel plates (disks). The upper plate is used to apply a rotation stress. The lower pallet supports and fixes the sample. This type of rheometer can apply stresses of 0.66 to 600 Pa and varying the frequency of at 40 Hz. In practice, these domains depend on the tested material. In the case of agar gel, we have been able to achieve the values (00 Pa, 40Hz) with the 8%concentration gel. Beyond these values (00 Pa, 40Hz) we noticed simple fractures. The gel is obtained by mixing the desired amount of the powder of the gel with water under a temperature of 70 to C for 10-15mins, and then the mixture was allowed to cool down to the ambient temperature for 30-60mins. Once the gel takes its solid state, samples are cut in cylindrical shape ( cm in diameter and mm thick). Fig. 1 shows a simplified diagram of the rheometer. The sample is placed between two pallets. The sample has the same diameter as the upper pallet. To achieve a rotary shear motion the lower surface of the gel must be fixed to the lower pallet of rheometer. This is ensured thanks to the adhesion of agar gel. * mchakroun1@yahoo.fr

2 The African Review of Physics (016) 11: Sample θ Free Plan Fixed plan Fig.1: Simplified diagram of the rheometer.1. Linearity test This test is conducted to measure the deformation of the sample as a function of the imposed stress. The behavior of the gel is linear if the relationship between deformation and stress is linear. Noting that for small strain, all materials have a linear behavior. In the case of a rotating shear, the deformation is given by the relationship ε (R/e).θ (where, e is thickness of the sample, R the sample radius, and θ represents the displacement angle)... Creep test The stress level imposed on the sample and the evolution of deformation was observed over time. Usually a viscoelastic material has a delayed deformation over time [1]..3. Oscillatory test The determination of the viscoelastic properties of the agar gel requires the establishment of vibration analysis. This type of testing is imposed on the sample an oscillating shear. The sinusoidal stress σ is out of phase with the strain ε [13] ε (ω) ε 0.cos (ωt) (1) σ (ω) σ 0.cos (ωt φ) ().3.1. Complex parameters In this test the complex shear modulus G * (ω) is defined by G * ( ω) σ( ωt) ε( ωt) σ0 ε0 e jϕ G ( ω) jg ( ω) (3) The real part of this complex modulus is the elastic modulus. It reflects the ability of the fluid to conserve energy in elastic form. The imaginary part is the loss modulus. It represents the viscous properties..3.. Linear viscoelasticity A material is linear viscoelastic if under stress (or deformed) it follows a kind of behavior law give by a σ 0 σ a1 b ε b 0 1 a ε σ b L ε.3.3. Mathematical modeling L (4) The viscoelastic behavior of the agar gel can be described conceptually as resulting from the combination of springs and shock absorbers. All associations of springs and dampers, as complex as they are, represent a linear viscoelastic behavior. From the association laws in series and in parallel, we can establish the rheological equation of each model (consisting of springs and dampers) and deduce the creep and relaxation functions in each particular case. The viscoelastic model used to simulate the mechanical behavior of agar gel is the Maxwell- Kelvin Voigt model. For constant stress σ 0 [13] the function of the creep compliance J (t) is J ε E E E 1 E t (t) 1 [1 exp( t)] exp( ) (5) σ0 E1E η E1 η 3. Results and Discussion 3.1. Results of the linearity test All tests were performed on several samples of the same concentration and reproduced several times. Fig. shows the results of agar gel linearity tests for %, 4% and 8% concentrations and 10 Hz frequency.

3 The African Review of Physics (016) 11: Note that this gel, present in different concentrations and low distortion ( 0.01), and with good linearity between the applied stress and the deformation. The gel is deformed less when its concentration is increased, which implies that its rigidity increases with the concentration. When the concentration is increased we can apply higher stresses. In our experiments we chose the stress value of 100 Pa to satisfy the condition that the gel with these different concentrations is in the linear viscoelastic region. Fig.: Agar gel linearity tests at different concentrations for T 0 0 C 3.. Result of creep test The temperature was kept constant at 0 0 C and the gel concentration was 8%. The sample is cylindrical, cm in diameter and mm thick. For a constant stress σ Pa, the function of the creep compliance J(t) of the Maxwell-Kelvin-Voigt model is given by Eqn. (5), which implies that the function of the deformation of this model is E E E 1 E t (t) 100 [ 1 [1 exp( ε t)] exp( ) ] (6) E1E η E1 η Fig. 3 shows the simulation result of the 8% agar gel creep test. It is noted that the deformation of the gel has an instantaneous elasticity followed by a delayed elasticity. The theoretical model parameters are calculated using MATLAB: - E Pa - E Pa - η Pa.s We note that we have a very good simulation of the experimental creep curve. This model allowed us to deduce the gel's mechanical parameters: - Instantaneous elasticity coefficient: E Pa - Delayed elasticity coefficient: E Pa - Coefficient of steady elasticity E o E1E E E 1 - And creep viscosity η Pa.s 3.3. Results of oscillatory tests In this case, the applied stress is sinusoidal (σ100 Pa). We present in Fig. 4 the measurement of the dynamic modulus G* of agar gel based on low frequencies (0.1-40Hz) for different concentrations and constant temperature 0 0 C. The curves are presented in semi logarithmic scale for frequencies. We also present the dynamic modulus of agar gel of human brain tissue and pork (biologically similar to that of humans) determined at low frequency. These results concerning the brain rigidity of tissue are among the few results found in the literature at low frequencies (<50 Hz). Note that the dynamic rigidity modulus decreases with increasing of the concentration of agar gel.

4 The African Review of Physics (016) 11: Fig.3: The creep Modeling test by the Maxwell -model -Kelvin Voigt Fig.4: Comparison of agar gel dynamic modulus by that of brain tissue at low frequencies Note that the dynamic modulus of agar gel with the different concentrations is of the same order of magnitude than the brain tissue. The 8% agar gel simulates well the results of Fallenchtain [8] and Thibault [9]. The % gel approaches the results obtained by Shuck [14]. It should be noted here that the mechanical properties of the brain matter depend on experimental conditions imposed by us, such as species, age, sample thickness, the heterogeneity and anisotropy of the brain, post mortem time and temperature. For this reason, the results presented by the authors in the literature are even in high frequencies quite different. 4. Conclusions Measurements made in creep and dynamic regime showed the linear viscoelastic properties of the agar gel at low frequency ( Hz) with stress of not more than 00 Pa. The gel has variable mechanical

5 The African Review of Physics (016) 11: properties depending on the concentration. A great choice of concentration allowed us to simulate a biological material such as brain matter. Brain matter is rarely studied in low frequency (<50 Hz). Most experiments on brain tissue are in a frequency domain between 50 and 600 Hz [15-1]. The 8% agar gel (at 0 0 C) simulates very well the dynamic modulus of pork and human brain matters to low frequencies. The mathematical model of Maxwell Kelvin Voigt type represents satisfactorily the viscoelastic behavior of this gel. The instantaneous coefficient of elasticity of this gel deduced by mathematical modeling is E Pa. This value is close to that measured by Galford [] on the human brain at 3.43 kpa. Also this value is in good agreement with those measured by other technical and elastography [3-5]. This gel can be used to simulate cerebral matter within the physical models of the human head. The results of mathematical modeling of creep test can be operated in finite element calculates code model of the human head. The agar gel offers us the opportunity to simulate mathematically and physically a wide range of biological soft tissues, such as muscle, lung, pancreas [6] and others. References [1] P. Coussot and J. L. Grossiord, Comprendre la Rhéologie, EDP Sciences (001). [] D Vaino, B. Aldaman and K. Pape, J. Crashworthiness, 191 (1997). [3] V. H. Kenner and W. J. Goldsmith, J. Biomechanics 6, 1 (1973). [4] Y. Yanagida, S. Fujiwara and Y. Mizoi, Forensic Science International 43, 159 (1989). [5] R. Willinger and D. Baumgartner, Int. J. Vehicle Design 31, 94 (003). [6] Dave W. A. Brands, Finite shear behavior of brain tissue under impact loading. Occupant protection and biomechanics in transportation, November 5-10, Oriando, Floria, USA (000). [7] R. Willinger, C. M. Kopp and D. Césari, New concept of countercoup lesion mechanism: modal analysis of a finite element model of the head, Proc. Int. IRCOBI Conf. on the Biomechanics of Impacts, p.83 (199). [8] G. T. Fallenstein, V. D. Hulce and J. W. Melvin, J. Biomechanics, 17 (1969). [9] K. L. Thibault and S. S. Margulies, Material Properties of the Developing Porcine Brain, IRCOBI International Conference on the Biomechanics of Impact, Dublin, Ireland (1996). [10] Dave W. A. Brand, Comparison of the dynamic behavior of brain tissue and two model materials, SAE Technical papers, p.10 (1999). [11] Difco Laboratories, Difco and BBL Manual of Microbiological Culture Media, Becton, Dickinson and Co., Sparks Maryland, p.696 (003). [1] G. Couarraze and J. L. Grossiord, Initiation à la rhéologie, Technique et documentation, Lavoisier, 3 ème édition (000). [13] Y. M. Hadad, Viscoelastic of engineering materials, ème édition, p.35 (00). [14] L. Z. Shuck and S. H. Advani, ASME J. Basic Engineering 94, 905 (197). [15] J. A. W. Van Dommelen, M. Hrapko and G. W. M. Peters, Mechanical Properties of Brain Tissue: Characterization and Constitutive Modeling, Springer Science, Mechanosensitivity of the Nervous System,, 49 (009). [16] K. B. Arbogast and S. S. Margulies, J. Biomech. 31, 801 (1998). [17] Zerrari Naoual, Caractérisation des tissus biologique mous par diffusion multiple de la lumière, Thèse de doctorat, Université Claude Bernard Lyon. Ecole doctorale Mega (014). [18] M. Chakroun, M. Zagrouba and I. Elloumi, J. Rhéologie 6, 67 (004). [19] M. Chakroun, M. H. Ben Ghozlen, I. Elloumi and S. Nicolle, Compte Rendu Physique, Elsevier 10, 36 (009). [0] Moez Chakroun, M. Hédi Ben Ghozlen, J. Mod. Phys. Scientific Research 3, 71 (01). [1] Moez Chakroun and Med Hédi Ben Ghozlen, Modélisation du module de cisaillement de la matière cérébrale, Revues Afrique Science ( 1(1), 110 Janvier (016). [] J. Galford and J. McElhaney, J. Biomechanics 3(), 11 (1970). [3] D. B. Plewes et al., JMRI 5, 733 (1995). [4] N. Bayly and E. H. Clayton, Measurement of Brain Biomechanics in vivo by Magnetic Resonance Imaging, Application of Imaging Techniques to Mechanics of Materials and Structures, Conference Proceedings of the Society for Experimental Mechanics Series 4, 117

6 The African Review of Physics (016) 11: (013). [5] R. Muthupillai et al., Science 69, 1854 (1995). [6] Nicolle L. Noguer and J. F. Palierne, J. Mechanical Behavior of Biomedical Materials 6, 90 (013). Received: May, 016 Accepted: September, 016