Biomedical Engineering – From Theory to Applications

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476 Biomedical Engineering – From Theory to Applications The spring constants ks, kA, and ka were estimated by the tensile test simulations such that the elastic energy generated in the mechano-cell equaled the strain energy WD obtained when the CM was modelled as a continuum. According to the theory of continuum mechanics, the strain energy WD is defined as Ne 1 D e 2 e1 T e e W A hεDε where Ne is the number of elements, Ae is the area of each element, and h is the thickness of the CM, e T = (Xe, Ye, XYe) is the strain vector of each element. D is the elastic modulus matrix under a plane strain condition. The parameters in eq. (16) were set to h = 0.5 m, elastic modulus of the CM ECM = 1000 Pa, and Poisson’s ratio v = 0.3 by reference to Feneberg et al. (2004) McGarry et al. (2004), and Mahaffy et al. (2004). Note that the elastic modulus and Poisson’s ratio appear in the elastic modulus matrix D. Energy W (pJ) 0.5 0.4 0.3 0.2 0.1 W D W s + W A 0 0 2 4 6 8 10 Cell deformation D (μm) Fig. 7. Elastic energy of the in-plane deformations (Ws + WA) stored in the mechano-cell (solid line) and the strain energy WD obtained when the CM was modelled as a continuum (dashed line). The spring constant of the bending spring kb was determined such that the bending energy Wb calculated from eq. (2) at the initial state of the cell equaled the bending energy WB analytically calculated (Wada and Kobayashi, 2003). Analytically, the bending energy WB of a sphere is given by 1 WB B C C dA 2 (16) 2 ( 1 2) (17) where B is the bending stiffness and C1 and C2 are the principal curvatures. Applying eq. (17) to the cell, allowing Ω to be CM and given that B = 2.010 -18 J (Zhelev et al., 1994) and C1 = C2 = 1/R0 (R0 = 10 m, initial radius of a cell), it follows that kb = 9.010 3 gm/s 2. The spring constant of the CSK kCSK was set to 1.510 6 g/s 2, based on the elastic modulus of an actin bundle (Deguchi et al., 2005). The CSKs were assumed to have a natural length when the cell was in its natural state. The CSKs were chosen randomly from all possible candidates of CSKs that were made by connecting two nodes on the CM. The spring
A Mechanical Cell Model and Its Application to Cellular Biomechanics constants of the volume elasticity (kV) were determined to assure cell incompressibility. Because no data is presently available for kn, it was determined that the load-deformation curves obtained by the simulation, fit the range of the experimental data. 3. Tensile tests 3.1 Tensile tests The mechanical behavior of a cell during a tensile test was simulated. The tensile test was simulated by fixing the nodes of CM at one side, while moving those at the opposite side in the direction of cell stretching. 3.2 Simulation results Figure 8 shows the deformation behaviour of a cell in the tensile test where a fibroblast is stretched, obtained by simulation of the model (left) and experimentally (right). Similar to the experimental data, the simulation showed that the cell and nucleus were elongated in the stretched direction. CSKs were randomly oriented prior to loading and were passively aligned in the stretched direction as the cell was stretched. Computational Experimental Nucleus Cell CSK Nucleus Cell 0 10 20 30 40 (µm) 0 10 20 30 40 (µm) Fig. 8. Snapshots of a cell during the tensile test simulation (left) and experimental system (right). The scale is indicated at the bottom of the figure. Load-deformation curves obtained from the simulation and experimental systems are presented in Fig. 9. Note that, in addition to the model with randomly oriented CSKs (Fig. 8), the data obtained from the models with parallel-oriented, oriented, and perpendicularly oriented CSKs, in addition to with no CSK are presented for comparison. The curve obtained from the simulation of the model with randomly oriented CSKs appeared to increase non-linearly. The curve of the model with randomly oriented CSKs lay within the variation of the experimentally obtained curves (simulation = 0.48 μN, experimental = 0.43- 1.24 μN at 20 μm cell deformation). Moreover, the curves obtained from the experiments were between the curve of the parallel-oriented model and that of perpendicularly oriented model. 477
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BIOMEDICAL ENGINEERING - FROM THEOR
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VI Contents Chapter 9 Targeted Magn
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X Preface stand-alone and readers c
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6 Biomedical search engine Scientif
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110 Method (Detection) Analyte Matr
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120 6. Conclusion Biomedical Engine
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150 5. Conclusions Biomedical Engin
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162 1 st phase : half year 4 2 nd p
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166 7. Innovative active training B
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172 12. Maturing projects’ story
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180 16. Acknowledgments Biomedical
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190 R* M M M M MR*M O O O O M O R*
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196 Fig. 9. Chemical structure of 5
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198 Relative Intensity (AU) Biomedi
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204 2. Preparation of IO nanopartic
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