MATHEMATICAL MODEL OF OUTER HAIR CELLS IN THE COCHLEA LI HAILONG NATIONAL UNIVERSITY OF SINGAPORE 2007 MATHEMATICAL MODEL OF OUTER HAIR CELLS IN THE COCHLEA LI HAILONG (B.ENG., M.ENG. XJTU) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgements Acknowledgements First of all, I would like to give my heartfelt gratitude to my supervisor Dr. Lim Kian Meng, for his invaluable guidance, support and encouragement throughout this entire research. His profound knowledge in mechanical dynamics and serious attitude towards academic research will benefit my whole life. I would like to thank Mr. He Xuefei and Dr. Lu Feng for the interesting and insightful discussion about vibration system. Special thanks to Dr. Wu Jiuhui for his sincere help and timely encouragement in the first two years of my research. I would also like to thank Li Mingzhou, Liu Guangyan, Zhou Lei, Tang Shan, Hu Yingping and Chen Yu, my best friends in Singapore, for the unforgettable happiness and hardship shared with me. During the four years of my research, their care and support deserve a lifetime memory. Finally, I would like to express my deepest gratitude and love to my parents and wife for their self-giving and continuous understanding and support. i Table of Contents Table of Contents Acknowledgements i Table of Contents ii Summary v List of Figures .vii List of Tables x 1. Introduction 1.1 Background .1 1.2 Purposes and Significance 1.3 Present Work .5 1.4 Organization of Thesis 2. Anatomy and Physiology of Ear .10 2.1 Anatomy of the Ear .10 2.2 Physiology of the Cochlea 12 2.3 Cochlear Mechanics 16 2.4 Physiology of Outer Hair Cell (OHC) 19 2.5 Summary .22 3. Mathematical Model of Outer Hair Cell .23 3.1 Literature Review 23 3.1.1 Quasi-static Models 23 3.1.2 Dynamic Models .26 3.2 Mathematical Formulation 27 3.2.1 Lateral Wall 28 3.2.2 Intracellular and Extracellular Fluids .33 3.2.3 Boundary Conditions 36 3.3 Parameter Determination 39 3.3.1 Quasi-static Axisymmetric Deformation 39 3.3.2 Iterative Method .42 ii Table of Contents 3.3.3 Code Validation 44 3.3.4 Results 45 3.3.5 OHC Length-dependent Properties of the Lateral Wall .46 3.4 Summary .49 4. Outer Hair Cell with Inviscid Flow 50 4.1 Literature Review 50 4.2 Parameters .52 4.3 Frequency Response by FDM .52 4.3.1 Equation Formulation .53 4.3.2 Results 55 4.4 Frequency Response by Coupled BEM/FDM 60 4.4.1 Equation Formulation .60 4.4.2 Results 65 4.5 OHC Resonant Frequency 66 4.5.1 OHC Length-dependent Resonant Frequency 67 4.5.2 Correlation of OHC Resonant Frequency with Cochlear Best Frequency .69 4.6 Summary .72 5. Outer Hair Cell with Viscous Flow 74 5.1 Literature Review .74 5.2 OHC Frequency Response 77 5.2.1 Formulation of Boundary Integral Equation (BIE) 77 5.2.2 Coupling of Fluid and Shell Equations .81 5.2.3 Code Validation 84 5.2.4 Mechanical Stimulation 85 5.2.5 Electrical Stimulation .86 5.3 Stereocilium Deflection 89 5.3.1 Model Description 89 5.3.2 Parameters 92 5.3.3 Electrically Induced Frequency Response 92 5.3.4 Stereocilium Deflection for Different Vibration Modes 95 iii Table of Contents 5.4 Summary .100 6. Outer Hair Cell Activity in the Cochlea 102 6.1 Literature Review 102 6.2 Model of Cochlear Partition 103 6.3 Parameters .107 6.3.1 Basilar Membrane .107 6.3.2 Outer Hair Cell .109 6.4 Forward Transduction .110 6.4.1 Amplitude .110 6.4.2 Phase .112 6.5 Results .113 6.5.1 BM Displacement Response .113 6.5.2 Parametric Study on OHC Forward Transduction .116 6.5.3 OHC Active Force 118 6.6 Summary .120 7. Conclusions .122 Bibliography 125 Appendix A Differential Operators Lij 142 Appendix B Differential Operators Lij′ 143 Appendix C Kernels of Inviscid Flow .144 Appendix D Stokslets of Oscillating Viscous Flow in Cylindrical Coordinates .146 Appendix E Kernels of Steady Viscous Flow 148 Publications .153 iv Summary Summary Previous studies on the outer hair cell (OHC) dynamics mainly focused on the axisymmetric vibration mode, and very little is known about the asymmetric vibration modes. In this thesis, a mathematical model of the OHC for different vibration modes is developed, including the coupling of the cell lateral wall with the intra- and extracellular fluids. The lateral wall is modeled as a cylindrical composite shell. For the fluids, two fluids models, inviscid and viscous flows, are used. Using the OHC model, the OHC electromechanical properties are determined by fitting available experimental measurements. These properties are found to be dependent on the OHC length. With the fluids modeled as an inviscid flow, the frequency responses for different vibration modes, together with the correlation of the OHC resonant frequencies with the cochlear best frequencies, are obtained using two different numerical methods. One method is an “all finite difference method (FDM)” where both shell and fluids equations are discretized by FDM. The other method is a “coupled boundary element/finite difference method (BEM/FDM)” where shell equation is discretized by FDM while fluid equation is discretized by BEM. The modeling results show that, at the basal turn of the cochlea, the OHC resonant frequency for the axisymmetric mode is close to the cochlear best frequency. At the apical turn, the resonant frequencies for the beam-bending mode and the pinched mode are closer to the cochlear best frequency. This important finding shows the correlation of OHC resonant frequencies with cochlear best frequencies. v Summary The inviscid flow model is also extended to a viscous flow model by including the fluid viscosity in the model. The numerical method is also an extension of the previous coupled BEM/FDM. Using BEM and taking advantage of the axisymmetric geometry, the present method is able to represent a three-dimensional oscillating viscous fluid problem with a one-dimensional domain. The results obtained show that, with the inclusion of viscosity, the frequency response is heavily damped, and the resonant frequency cannot be observed. Using a simple kinematic model of the organ of Corti, the contributions of the first two vibration modes to the streocilium deflection are analyzed. Besides the axisymmetric mode, the beam-bending mode may contribute to streocilium deflection over the hearing range. This contribution is comparable to that of the axisymmetric mode at the apical turn of the cochlea, but it becomes insignificant at the basal turn. The result is new to the literature on models of the organ of Corti, and it contributes to our knowledge of the dynamics in the cochlea. Finally, a feedback model of the cochlear partition is developed to obtain the OHC activity in the cochlea. Through comparison of the responses in the passive and active cochlear models, the OHC at the basal turn appears to contribute its active force to enhance the basilar membrane response, providing a positive feedback in the cochlea, while the OHC at the apical turn tends to contribute its active force to suppress the basilar membrane response, providing a negative feedback in the cochlea. Also, the amplification factor in the active cochlear model is found to be sensitive to the amplitude and phase angle of transfer function TF in the OHC forward transduction process. These findings are important to our understanding of OHC active roles played in the cochlea. vi List of Figures List of Figures Figure 2.1 Cross section of the human ear 10 Figure 2.2 Schematic drawing of the uncoiled cochlea 12 Figure 2.3 Drawing of the cross section of one cochlear turn 13 Figure 2.4 Drawing of the anatomy of the organ of Corti 15 Figure 2.5 Schematic drawings of the OHC and its lateral wall .19 Figure 2.6 OHC electromotiltiy and its sensitivity as a function of transmembrane voltage .21 Figure 3.1 Notations and positive directions of force and moment resultants of the cylindrical shell .31 Figure 3.2 Resultant stiffness modulus and Poisson’s ratio of the cortical lattice against the length of the OHC with large Poisson’s ratio of the plasma membrane (νP =0.9) .47 Figure 3.3 Resultant stiffness modulus and Poisson’s ratio of the cortical lattice against the length of the OHC with small Poisson’s ratio of the plasma membrane (νP=0.5) 48 Figure 4.1 Applied force on the circumference at the free end of the OHC to excite various vibration modes in circumferential direction .51 Figure 4.2 OHC deformation shapes at frequency 2000Hz for different vibration modes (k=0, 1, and 3) 56 Figure 4.3 OHC displacement responses at frequency 2000Hz for different vibration modes 57 Figure 4.4 Frequency response of the OHC with only intracellular fluid for axisymmetric mode (k=0) and beam-bending mode (k=1) 58 Figure 4.5 Frequency response of the OHC with both intracellular and extracelluar fluids for axisymmetric mode (k=0) and beam-bending mode (k=1) 58 Figure 4.6 Frequency response of the OHC in the case of inviscid flow for axisymmetric mode (k=0) and beam-bending mode (k =1) .65 Figure 4.7 Comparison of the computational time for the couple boundary vii List of Figures element/finite difference method and the “all finite difference method” for axisymmetric mode (k=0) at the frequency 2000Hz .66 Figure 4.8 Plots of OHC resonant frequency against the cell length for the first three vibration modes (k=0, and 2) 67 Figure 4.9 Fitted curves of the OHC resonant frequency against cell length for axisymmetic mode (k=0) and beam-bending mode (k=1) .68 Figure 4.10 Comparison of the first resonant frequency of the OHC to the best frequency of the cochlea (Pujol et al., 1992) for axisymmetric mode (k=0), beambending mode (k=1) and pinched mode (k=2) .70 Figure 5.1 Validation of the results obtained from present OHC model by using the modeling results of Tolomeo and Steele (1998) 84 Figure 5.2 Frequency responses (amplitude and phase) of the OHC with the length of 60 μm under mechanical stimulation, for axisymmetric mode (k=0) and beam-bending mode (k=1) .86 Figure 5.3 Frequency responses (amplitude and phase) of the OHC with the length of 60 μm under electrical stimulation, for axisymmetric mode (k=0) and beam-bending mode (k=1) .87 Figure 5.4 Comparison of the results from present OHC model with reported experimental and numerical results for cell length L= 60 μm 88 Figure 5.5 Kinematic model of the stereocilium deflection due to the OHC axisymmetric mode (k=0) and beam-bending mode (k=1) 90 Figure 5.6 Frequency response of the isolated OHC (KRL=0N/m) for axisymmetric mode (k=0) and beam-bending mode (k=1) (a) L=30 μm, and (b) L=60 μm 93 Figure 5.7 Frequency response of the constrained OHC (KRL=0.05 N/m) for axisymmetric mode (k=0) and beam-bending mode (k=1) (a) L=30 μm, and (b) L=60 μm .94 Figure 5.8 Plot of the parameter λ against angle α at the basal and apical turns of the cochlea 95 Figure 5.9 Stereocilium deflection resulted by axisymmetric mode (k=0) and beambending mode (k=1) for the OHC with the length of 30 μm and 60 μm 98 Figure 6.1 Model of the cochlear partition .104 Figure 6.2 Flow chart of the feedback system in the cochlea .105 Figure 6.3 Feedback model of active cochlea .106 viii Bibliography Sit, P.S., Spector, A.A., Lue, A.J., Popel, A.S. and Brownell, W.E. 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Lawrence Erlbaum Associates, Mahwah, New Jersey. 2002 141 Appendix A: Differential Operators Lij Appendix A Differential Operators Lij L11 = A11 L12 = ( L13 = ∂ A66 ∂ + ∂x R ∂θ A12 + A66 B12 + B66 ∂ ) + R R2 ∂x∂θ A12 ∂ B12 + B66 ∂ ∂3 − − B 11 R ∂x R2 ∂x∂θ ∂x L21 = ( A12 + A66 B12 + B66 ∂ + ) R R2 ∂x∂θ B66 D66 ∂ A22 B22 D22 ∂ + + ) + ( A + + 2) 66 R2 R3 R ∂θ R R ∂x B + B66 D12 + D66 A B B22 D22 ∂ ∂ ∂3 L23 = ( 222 + 223 ) − ( 12 + ) − ( + 4) R R ∂θ R R2 ∂x ∂θ R3 R ∂θ 3 ⎛ A ∂ B12 + B66 ∂ ∂ ⎞ L31 = −⎜⎜ 12 − − B11 ⎟⎟ 2 R ∂x∂θ ∂x ⎠ ⎝ R ∂x L22 = ( ⎡ A B + B66 D12 + D66 B B22 D22 ∂ ⎤ ∂ ∂3 − ( 12 + L32 = − ⎢( 222 + 223 ) − + ) 3⎥ ) ( ∂x ∂θ R ∂θ R R2 R3 R ∂θ ⎦ ⎣ R 2( D12 + D66 ) ∂ A ∂ D22 ∂ 2B ∂ 2B ∂ L33 = − 222 + 12 + 322 − − D − 11 R R ∂x R ∂θ R2 ∂x ∂θ ∂x R ∂θ 142 Appendix B: Differential Operators Lij′ Appendix B Differential Operators Lij′ Ω = (ρ1h1 + ρ h2 )ω ⎡⎛ A ⎞ d2 ⎤ L11′ = ⎢⎜ Ω − k 662 ⎟ + A11 ⎥ R ⎠ dx ⎦ ⎣⎝ L12′ = k ( A12 + A66 B12 + B66 d ) + R R2 dx ⎡⎛ A B + 2B ⎞ d d3 ⎤ L13′ = ⎢⎜ 12 + k 12 66 ⎟ − B11 ⎥ R dx ⎦ ⎠ dx ⎣⎝ R ′ = −k ( L21 A12 + A66 B12 + B66 d ) + R R2 dx ⎡ 2B D d2 ⎤ A 2B D ′ = ⎢Ω − k ( 222 + 322 + 224 ) + ( A66 + 66 + 662 ) ⎥ L22 R R R R R dx ⎦ ⎣ ⎡ B + B66 D12 + D66 d ⎤ A B B D ′ = ⎢− k ( 222 + 223 ) − k ( 223 + 224 ) + k ( 12 + L23 ) 2⎥ R R R R R R2 dx ⎦ ⎣ ⎡⎛ A B + 2B ⎞ d d3 ⎤ ′ = − ⎢⎜ 12 + k 12 66 ⎟ − B11 ⎥ L31 R dx ⎦ ⎠ dx ⎣⎝ R ⎡ B + B66 D12 + D66 d ⎤ A B B D ′ = ⎢− k ( 222 + 223 ) − k ( 223 + 224 ) + k ( 12 ) 2⎥ + L32 R R2 dx ⎦ R R R R ⎣ ⎡⎛ A22 d4 ⎤ 2 B22 D22 ⎞ ⎛ B12 2( D12 + D66 ) ⎞ d ′ +k L33 = ⎢⎜ Ω − − k −k ⎟ − D11 ⎥ ⎟+⎜ dx ⎦ R4 ⎠ ⎝ R R2 R3 R ⎠ dx ⎣⎝ 143 Appendix C: Kernels of Inviscid Flow Appendix C Kernels of Inviscid Flow Complete elliptic Integrals of the first and second kinds are ⎛ π⎞ π K = K ⎜ m, ⎟ = ∫ ⎝ 2⎠ ⎛ π⎞ π E = E ⎜ m, ⎟ = ∫ ⎝ 2⎠ ) dα (1 − m sin α (1 − m sin α dα ) Variables C , D and m are C = ( R p + rQ ) + ( X p − xQ ) D = (R p − rQ ) + (X p − xQ ) m= R p rQ ( R p + rQ ) + ( X p − xQ ) The kernels in equations (4.16) and (4.17) are given by G0 = ⎛ π⎞ K ⎜ m, ⎟ πC ⎝ ⎠ ∂G0 ⎛ π ⎞ R p − rQ + (X p − xQ ) ⎛ π ⎞ X p − xQ ⎛ π⎞ nr K ⎜ m, ⎟ + nr E ⎜ m, ⎟ + n x E ⎜ m, ⎟ r =− 2πCrQ 2πCDrQ πCD ∂n ⎝ 2⎠ ⎝ 2⎠ ⎝ 2⎠ G1 = − ⎛ π⎞ ⎡ ⎛ π⎞ ⎛ π ⎞⎤ K ⎜ m, ⎟ + K ⎜ m, ⎟ − E ⎜ m, ⎟⎥ ⎢ πC ⎝ ⎠ m πC ⎣ ⎝ ⎠ ⎝ ⎠⎦ ∂G0 ⎡ ⎛ ∂m ∂G1 ∂m ⎞⎤ ⎡ ⎛ π ⎞ ⎛ π ⎞⎤ nr + n x ⎟⎥ K ⎜ m, ⎟ − E ⎜ m, ⎟⎥ + . r = − r + ⎢− ⎜ ⎢ ∂n ∂n ⎣ m ⎝ ∂r ∂x ⎠⎦ πC ⎣ ⎝ ⎠ ⎝ ⎠⎦ ⎡ ⎛ ∂C ∂C ⎞⎤ ⎡ ⎛ π ⎞ ⎛ π ⎞⎤ nr + n x ⎟⎥ ⎢ K ⎜ m, ⎟ − E ⎜ m, ⎟⎥ + . − ⎢ ⎜ ∂x ⎠⎦ ⎣ ⎝ ⎠ m ⎣ πC ⎝ ∂r ⎝ ⎠⎦ ⎡ m ⎛ ∂m ∂m ⎞ ⎛ π ⎞⎤ nr + n x ⎟ E ⎜ m, ⎟ ⎢ 2 ⎜ ∂x ⎠ ⎝ ⎠⎥⎦ m πC ⎣1 − m ⎝ ∂r 4 G2 = − G0 − G1 + G1 3 k2 144 Appendix C: Kernels of Inviscid Flow ∂G2 ∂m ⎞⎤ ∂G0 ∂G1 ∂G1 ⎡ ⎛ ∂m nr + n x ⎟ G1 r =− r − r + r + ⎢− ⎜ ∂n ∂x ⎠⎥⎦ ∂n ∂n 3m ∂n ⎣ m ⎝ ∂r where n x and nr are axial and radial components of the unit outward normal of the fluid r domain n (either intracellular or extracellular fluids). The circumferential component of r n is zero due to the axisymmetric shape of the domain. 145 Appendix D: Stokslets of Oscillating Viscous Flow in Cylindrical Coordinates Appendix D Stokslets of Oscillating Viscous Flow in Cylindrical Coordinates The variable x is defined as: x = xQ − X p xQ − X p ∂r rQ − R p cosθ nr + nx r= ∂n r r ⎡ cosθ (rQ − R p cosθ ) (rQ cosθ − R p )⎤ + B(λr ) ⎢ A(λr ) ⎥ 8πμ ⎣ r r r r ⎦ ⎡ sin θ R p sin θ (rQ cosθ − R p )⎤ = + B(λr ) ⎢− A(λr ) ⎥ 8πμ ⎣ r r r r ⎦ Grrλ = Gθλr ⎡ x (rQ cosθ − R p )⎤ ⎢ B (λr ) ⎥ rr r 8πμ ⎣ ⎦ ⎡ sin θ (rQ − R p cosθ ) rQ sin θ ⎤ = + B (λr ) ⎢ A(λr ) ⎥ r r r r ⎦ 8πμ ⎣ G xrλ = Grλθ Gθθλ = ⎡ cos θ rQ sin θ R p sin θ ⎤ + B (λr ) ⎢ A(λr ) ⎥ r r r r 8πμ ⎣ ⎦ G xλθ = ⎡ x rQ sin θ ⎤ ⎢ B (λr ) ⎥ rr r ⎦ 8πμ ⎣ Grxλ = ⎡ (rQ − R p cosθ ) x ⎤ ⎢ B (λr ) ⎥ r r r⎦ 8πμ ⎣ Gθλx = ⎡ R p sin θ x ⎤ ⎢ B (λr ) ⎥ r r r⎦ 8πμ ⎣ ⎡ 1⎛ x ⎞ ⎤ Gxx = ⎢ A(λr ) + B(λr ) ⎜ ⎟ ⎥ 8πμ ⎣⎢ r r ⎝ r ⎠ ⎦⎥ λ2 146 Appendix D: Trrλ = 8π Tθλr = 8π Txrλ 8π Tθθλ = 8π Txλθ = 8π Txxλ 2(rQ − R p cosθ )sin θ ⎧ 2rQ sin θ ⎫ nr [1 − B(λr )] − nr C (λr ) − .⎪ ⎪⎪− 3 ⎪ r r ⎨ ⎬ ⎪ sin θ ∂r C (λr ) + 2(rQ − R p cosθ )rQ sin θ ∂r [D(λr ) − 3] ⎪ r ⎪⎩ r ∂nr ⎪⎭ r4 ∂n ⎫⎪ 2rQ R p sin θ ∂r cosθ ∂r ⎪⎧ R p sin θ ( ) ( ) λ λ C r + n C r − − r r [D(λr ) − 3]⎬ ⎨ r ∂n ∂n r r r ⎪⎩ ⎪⎭ x rQ sin θ ∂r ⎧ 2rQ sin θ ⎫ x sin θ n x [1 − B(λr )] − nr C (λr ) + r [D(λr ) − 3]⎬ ⎨− 3 ∂n r r r ⎩ ⎭ 2(rQ − R p cosθ ) 2(rQ − R p cosθ )x ∂r ⎫ ⎧ 2x n x C (λr ) + r [D(λr ) − 3]⎬ ⎨− nr [1 − B(λr )] − r r ∂n ⎭ ⎩ r R p x sin θ ∂r ⎫ ⎧ R p sin θ = n x C (λr ) + r [D(λr ) − 3]⎬ ⎨− ∂n 8π ⎩ r r ⎭ ⎫ ⎧ 2x ⎡ ∂r 2( x ) ∂r ⎤ = r [D(λr ) − 3]⎬ ⎨− n x [1 − B(λr )] − ⎢r r + x n x ⎥C (λr ) + 8π ⎩ r r ⎣ ∂n r ∂n ⎦ ⎭ Trxλ = Tθλx 2(rQ − R p cosθ )cosθ ⎫ ⎧ 2(rQ cosθ − R p ) nr [1 − B(λr )] − nr C (λr ) − .⎪ ⎪⎪− 3 ⎪ r r ⎬ ⎨ ⎪ cosθ ∂r C (λr ) + 2(rQ − R p cosθ )(rQ cosθ − R p ) ∂r [D(λr ) − 3] ⎪ r ⎪⎭ ⎪⎩ r ∂nr ∂n r4 ⎧ R p sin θ cosθ ⎫ sin θ ∂r nr C (λr ) − r C (λr ) + .⎪ ⎪⎪− ⎪ ∂n r r ⎨ ⎬ ( ) ⎪ rQ cosθ − R p R p sin θ ∂r [D(λr ) − 3] ⎪ r ⎪⎩ ⎪⎭ ∂n r4 ⎧ 2(rQ cosθ − R p ) ⎫ x cosθ n x [1 − B(λr )] − nr C (λr ) + .⎪ − ⎪ 3 ⎪ ⎪ r r = ⎨ ⎬ 8π ⎪ x (rQ cosθ − R p ) ∂r ⎪ r [D(λr ) − 3] ⎪⎩ ⎪⎭ r ∂n Trλθ = Stokslets of Oscillating Viscous Flow in Cylindrical Coordinates 8π 147 Appendix E: Kernels of Steady Viscous Flow Appendix E Kernels of Steady Viscous Flow The variables I αβ and Η αβ are defined as follows. π I αβ = ∫ −π cos β θ dθ rα π Η αβ = ∫ Η 61 = Η 62 = ( 1− k sin α ) β +1 15m − 34m + 27 m − ⎛ π ⎞ 23m − 23m + ⎛ π ⎞ E ⎜ m, ⎟ K ⎜ m, ⎟ + 15m 15m ⎝ 2⎠ ⎝ 2⎠ (8 − 9m )(1 − m ) K ⎛⎜ m, π ⎞⎟ + − + 13m 3m Η 64 = − I10 = cosα α ⎝ 2⎠ 3m − 3m ⎛ π⎞ E ⎜ m, ⎟ ⎝ 2⎠ − m − m ⎛ π ⎞ − 3m − 2m ⎛ π ⎞ K ⎜ m, ⎟ + E ⎜ m, ⎟ 3m 3m ⎝ 2⎠ ⎝ 2⎠ ⎛ π⎞ K ⎜ m, ⎟ C ⎝ 2⎠ ⎡ − m ⎛ π ⎞ ⎛ π ⎞⎤ I 11 = − ⎢− K ⎜ m, ⎟ + E ⎜ m, ⎟⎥ + I 10 C ⎣ m2 ⎝ ⎠ m ⎝ ⎠⎦ ( )( ) ( ) 16 ⎡ − 3m − m ⎛ π ⎞ 2m − ⎛ π ⎞⎤ I 12 = ⎢ K ⎜ m, ⎟ + E ⎜ m, ⎟⎥ + I11 − I10 C⎣ 3m 3m ⎝ 2⎠ ⎝ ⎠⎦ I 13 = − I 30 = ⎛ π⎞ E ⎜ m, ⎟ C (1 − m ) ⎝ ⎠ I 31 = − I 32 = 32 Η 61 + 3I12 − 3I 11 + I10 C ⎡ ⎛ π ⎞ ⎛ π ⎞⎤ K ⎜ m, ⎟ − E ⎜ m, ⎟ + I 30 C ⎢⎣ m ⎝ ⎠ m ⎝ ⎠⎥⎦ 16 C3 ( ) ⎡ − m ⎛ π ⎞ − m ⎛ π ⎞⎤ K ⎜ m, ⎟ + E ⎜ m, ⎟⎥ + I 31 − I 30 ⎢− m4 m4 ⎝ 2⎠ ⎝ ⎠⎦ ⎣ 148 Appendix E: I 33 = − 32 Η 62 + 3I 32 − 3I 31 + I 30 C3 [ ) ] ( 64 Η 61 − − m Η 62 + I 33 − I 32 + I 31 − I 30 C m I 34 = I 50 Kernels of Steady Viscous Flow = C I 51 = − I 52 = ) C5 ) ( ) ⎡ 2m − ⎛ π⎞ ⎛ π ⎞⎤ + K m , E ⎜ m, ⎟⎥ + I 50 ⎜ ⎟ ⎢ 2 ⎝ ⎠⎦ ⎣ 3m ⎝ ⎠ 3m − m ( ) ( ) 16 ⎡ + m ⎛ π ⎞ + m ⎛ π ⎞⎤ K ⎜ m, ⎟ − E ⎜ m, ⎟⎥ + I 51 − I 50 ⎢ 3m C ⎣ 3m ⎝ 2⎠ ⎝ ⎠⎦ I 53 = − I 54 = ( ( ⎡ ⎛ π ⎞ 2−m ⎛ π ⎞⎤ E ⎜ m, ⎟⎥ ⎢− K ⎜ m, ⎟ + 2 ⎝ ⎠ 1− m ⎝ ⎠⎦ − m ⎣ 32 Η 64 + 3I 52 − 3I 51 + I 50 C5 [ ) ] ( 64 Η 62 − − k Η 64 + I 53 − I 52 + I 51 − I 50 C 5m2 I 55 = − [ ( ) ( ] ) 128 Η 61 − − k Η 62 + − k Η 64 + I 54 − 10 I 53 + 10 I 52 − 5I 51 + I 50 C m The integrals of the kernels Gij and Tij are given by: k=0: π ∫πG − rr dθ = [I 8πμ dθ = π ∫πG − xr 8πμ [xr I Q 31 − π ∫πG − rx dθ = xx dθ = π ∫πG − π 8πμ π − xr Q 30 10 − x R p I 30 − x R p I 31 ] ] + rQ R p I 30 − rQ R p I 32 [xr I ] ] ] − (2r + x I 30 [ [ {[− xr R I dθ = − 4π ∫−π Trr dθ = − 4π ∫ πT 11 [I 8πμ ) − rQ R p I 32 + rQ + R p I 31 − rQ R p I 30 11 ∫ π Gθθ dθ = 8πμ [I π ( ) ( ) ) ] )I − xr R I ]n + [x r I ] 3 ⎧⎪ rQ R p I 53 ⎫ Q R p + R p I 52 + 2rQ R p + rQ I 51 − rQ R p I 50 nr + .⎪ ⎨ ⎬ 2 ⎪⎩ − x rQ R p I 52 + x rQ + R p I 51 − x rQ R p I 50 n x ⎪⎭ Q p 52 ( + x (r Q + Rp 51 Q p 50 r Q 51 ] } − x R p I 50 n x 149 Appendix E: Kernels of Steady Viscous Flow [ ∫−π Tθθ dθ = − 4π π ∫ πT rx − π ∫ πT − xx ] ⎧⎪ rQ R p I 53 − rQ R p I 52 − rQ R p I 51 + rQ R p I 50 nr + .⎫⎪ ⎨ ⎬ ⎪⎩ − x rQ R p I 52 + x rQ R p I 50 n x ⎪⎭ π [ ] {[ ] [ ] } dθ = − 2 x rQ I 50 − x rQ R p I 51 + x R p I 52 nr + x rQ I 50 − x R p I 51 n x 4π dθ = − − x R p I 51 + x rQ I 50 nr + x I 50 n x 4π {[ ] [ ] } k=1: π ∫πG − π ∫ π Gθ r − π ∫πG [I 8πμ ( cosθdθ = rr [− I 8πμ sin θdθ = ] + I12 + rQ R p I 31 − rQ R p I 33 − R p I 30 + R p I 32 10 [ cos θdθ = ) − rQ R p I 33 + rQ + R p I 32 − rQ R p I 31 12 ] x rQ I 32 − x R p I 31 8πμ π 2 − ∫ Grθ sin θdθ = − I10 + I12 + rQ R p I 31 − rQ R p I 33 − rQ I 30 + rQ I 32 −π 8πμ − xr [ ∫ π Gθθ cosθdθ = 8πμ [I π − π −π 8πμ − ∫ G xθ sin θdθ = π ∫πG − rx π ∫ π Gθ x sin θdθ = xx cosθdθ = − π ∫πG − cos θdθ = π 8πμ ∫π − Q 32 − x rQ I 30 ] [xR I − x R p I 32 ] Q 31 p 30 [I 8πμ 11 + x I 31 Tθr π xr ] ] ] ] [ ( [ ( ⎧⎪[r R I − (r R sin θdθ = − ⎨ 4π ⎪[− x r R I + x R ⎩ {[− xr R I + x (r cosθdθ = − 4π ) ( ) )I + (r ] ) ⎧⎪ rQ R p I 54 − 2rQ R p + R p I 53 + 2rQ R p + rQ I 52 − rQ R p I 51 nr + .⎫⎪ ⎨ ⎬ 2 ⎪⎩ − x rQ R p I 53 + x rQ + R p I 52 − x rQ R p I 51 n x ⎪⎭ Q ] ) ] ] p 54 Q p 53 R p + R p I 51 − rQ R p I 50 nr + .⎫⎪ ⎬ 2 ⎪⎭ p I 52 + x rQ R p I 51 − x R p I 50 n x Q p 53 Q ∫ πT − [xr I ∫−π Trr cosθdθ = − 4π π + rQ R p I 31 − rQ R p I 33 − xˆR p I 32 8πμ 12 [xˆr I ] Q p + Rp 53 ) Q ] [ ] } + R p I 52 − x rQ R p I 51 nr + x rQ I 52 − x R p I 51 n x 150 Appendix E: Kernels of Steady Viscous Flow ( ) ⎧⎡− rQ R p I 54 + 2rQ R p I 53 + rQ R p − rQ I 52 − .⎤ ⎫ ⎪ ⎪⎪ ⎢ ⎥ . + n π r ⎪ ⎢⎣2rQ R p I 51 + rQ I 50 ⎥⎦ − ∫ Trθ sin θdθ = ⎨ ⎬ −π 4π ⎪ ⎪ 2 ⎪⎩ x rQ R p I 53 − x rQ I 52 − x rQ R p I 51 + x rQ I 50 n x ⎪⎭ 2 2 π ⎧⎪ rQ R p I 54 − rQ R p I 53 − rQ R p I 52 + rQ R p I 51 nr + .⎫⎪ ⎬ ∫−π Tθθ cos θdθ = − 4π ⎨⎪ − x r R I + xr R I n ⎪⎭ Q p 53 Q p 51 x ⎩ 2 π ⎧⎪ x rQ R p I 53 − x rQ I 52 − x rQ R p I 51 + x rQ I 50 nr + .⎫⎪ − ∫ Txθ sin θdθ = ⎨ ⎬ −π 4π ⎪ − x rQ I 52 + x rQ I 50 n x ⎪⎭ ⎩ π 2 2 ∫−π Trx cos θdθ = − 4π xrQ I 51 − xrQ R p I 52 + x R p I 53 nr + x rQ I 51 − x R p I 52 nx [ [ ] [ [ ] [ ] ] {[ π ∫−π Tθx sin θdθ = − 4π π ∫ πT − xx cosθdθ = − ] ] [ [ ] } ] ⎧⎪ x R p I 53 − x rQ R p I 52 − x R p I 51 + x rQ R p I 50 nr + .⎫⎪ ⎨ ⎬ 2 ⎪⎩ − x R p I 52 + x R p I 50 n x ⎪⎭ [ ] {[ ] [ ] } − x R p I 52 + x rQ I 51 nr + x I 51 n x 4π k=2: ⎤ ⎡(2 I13 − I11 ) − rQ R p (2 I 34 − I 32 ) + . ⎢ ⎥ ∫−π 8πμ ⎢⎣ rQ + R p (2 I 33 − I 31 ) − rQ R p (2 I 32 − I 30 )⎥⎦ π 2 ∫−π Gθr sin 2θdθ = 8πμ − I11 + I13 + rQ R p I 32 − rQ R p I 34 − R p I 31 + R p I 33 π ∫−π Gxr cos 2θdθ = 8πμ x rQ (2I 33 − I 31 ) − x R p (2I 32 − I 30 ) π 2 − ∫ Grθ sin 2θdθ = − I11 + I13 + rQ R p I 32 − rQ R p I 34 − rQ I 31 + rQ I 33 −π 8πμ π ∫−π Gθθ cos 2θdθ = 8πμ (2I13 − I11 ) + rQ R p (2I 32 − I 30 ) − rQ R p (2I 34 − I 32 ) π − ∫ G xθ sin 2θdθ = x rQ I 33 − x rQ I 31 −π 8πμ π ∫−π Grx cos 2θdθ = 8πμ x rQ (2 I 32 − I 30 ) − xR p (2 I 33 − I 31 ) π ∫−π Gθx sin 2θdθ = 8πμ xR p I 31 − x R p I 33 π ∫−π Gxx cos 2θdθ = 8πμ (2I12 − I10 ) + x (2I 32 − I 30 ) π Grr cos 2θdθ = ( ) [ ] [ ] [ [ ] [ ] [ [ ] ] ] [ ] 151 Appendix E: Kernels of Steady Viscous Flow ( ) ⎫ ⎧⎡rQ R p (2 I 55 − I 53 ) − 2rQ R p + R p (2 I 54 − I 52 ) + .⎤ ⎪ ⎪⎢ ⎥ n . + r ⎪⎪ ⎪ ⎢ ⎥ ( ) ( ) r R r I I r R I I 2 + − − − π ⎪⎣ Q p Q Q p 53 51 52 50 ⎦ T d cos θ θ = − ⎬ ⎨ ∫−π rr 4π ⎪⎡− x r R (2 I − I ) + x r + R (2 I − I ) − .⎤ ⎪ 54 52 53 51 Q p Q p ⎥ nx ⎪ ⎪⎢ ⎥⎦ ⎪⎭ ⎪⎩⎢⎣ x rQ R p (2 I 52 − I 50 ) ⎫ ⎧⎡rQ R p I 55 − rQ R p + R p I 54 + rQ R p + R p I 52 − .⎤ ⎪⎪ ⎪ ⎢ ⎥ . n + π r ⎪ ⎢ ⎥ sin T d θ θ = − r R I ⎬ ⎨ θ r ∫−π ⎦ 4π ⎪⎣ Q p 51 ⎪ 2 ⎪⎭ ⎪⎩ − x rQ R p I 54 + x R p I 53 + x rQ R p I 52 − x R p I 51 n x ( ) ( ( ) ) [ ( ( ) ] ) ⎫ ⎧⎡− x rQ R p (2 I 54 − I 52 ) + x rQ + R p (2 I 53 − I 51 ) − .⎤ ⎪⎪ ⎪ n . + ⎥ ⎢ π r ⎪ ( ) x r R I I − T cos d θ θ = − ⎥ ⎢ ⎬ ⎨ 52 50 Q p xr ∫−π ⎦ 4π ⎪⎣ ⎪ 2 ⎭⎪ ⎩⎪ x rQ (2 I 53 − I 51 ) − x R p (2 I 52 − I 50 ) n x [ ( ] ) ⎫ ⎧⎡− rQ R p I 55 + 2rQ R p I 54 + rQ R p − rQ I 53 − .⎤ ⎪⎪ ⎪ ⎢ ⎥ . n + π r ⎪ ⎢⎣2rQ R p I 52 + rQ I 51 ⎥⎦ − ∫ Trθ sin 2θdθ = ⎬ ⎨ −π 4π ⎪ ⎪ 2 ⎪⎭ ⎪⎩ x rQ R p I 54 − x rQ I 53 − x rQ R p I 52 + x rQ I 51 n x ⎧⎡rQ R p (2 I 55 − I 53 ) − rQ R p (2 I 54 − I 52 ) + .⎤ ⎫ ⎪ ⎪⎪ ⎢ ⎥ n . + π ⎪ r 2 T cos d θ θ = − ⎥ ⎢ ( ) ( ) r R I I r R I I − − + − ⎨ ⎬ θθ ∫−π Q p 53 51 52 50 ⎦ 4π ⎪⎣ Q p ⎪ ⎩⎪ − x rQ R p (2 I 54 − I 52 ) + x rQ R p (2 I 52 − I 50 ) n x ⎭⎪ [ ] [ ] [ ] ⎧⎪ x rQ R p I 54 − x rQ I 53 − x rQ R p I 52 + x rQ I 51 nr + .⎫⎪ ⎬ ⎨ 2 ⎪⎭ ⎪⎩ − x rQ I 53 + x rQ I 51 n x 2 π ⎧⎪ x rQ (2 I 52 − I 50 ) − x rQ R p (2 I 53 − I 51 ) + x R p (2 I 54 − I 52 ) nr + .⎫⎪ θ θ T cos d = − ⎨ ⎬ rx ∫−π 4π ⎪ x rQ (2 I 52 − I 50 ) − x R p (2 I 53 − I 51 ) n x ⎪⎭ ⎩ 2 π ⎧⎪ x R p I 54 − x rQ R p I 53 − x R p I 52 + x rQ R p I 51 nr + .⎫⎪ θ θ sin T d = − ⎨ ⎬ θ x ∫−π 4π ⎪ − x R p I 53 + x R p I 51 n x ⎪⎭ ⎩ π 2 ∫−π Txx cos 2θdθ = − 4π − x R p (2 I 53 − I 51 ) + x rQ (2 I 52 − I 50 ) nr + x (2I 52 − I 50 ) nx π − ∫ Txθ sin 2θdθ = −π 4π [ [ ] [ [ [ {[ ] ] ] ] ] [ ] } 152 Publications Publications Lim, K.M. and Li, H.L. A two-layer outer hair cell model with orthotropic piezoelectric properties: Correlation of cell resonant frequencies with tuning in the cochlea. Journal of Biomechanics, 40, pp. 1362-1371. 2007. Lim, K.M. and Li, H.L. A coupled boundary element/finite difference method for fluid-structure interaction with application to dynamic analysis of outer hair cells. Computers and Structures, 85, pp. 911-922. 2007. Li, H.L. and Lim, K.M. Contribution of outer hair cell bending to stereocilium deflection in the cochlea. Hearing Research, 232, pp. 20-28. 2007. 153 [...]... property of the outer hair cell 22 Chapter 3: Mathematical Model of Outer Hair Cell Chapter 3 Mathematical Model of Outer Hair Cell The mathematical model of the outer hair cell is presented in this chapter Firstly, a brief literature review on previously developed mathematical models of the outer hair cell is given The detailed mathematical formulation used in the present model is then described Lastly,... cochlea By including the stiff constraint of the reticular lamina on the OHC, the relationship between the OHC stereocilium deflection and its first two vibration modes is discussed The OHC is finally integrated into a simple model of the cochlear partition and the OHC active roles played in the cochlea are studied The OHC model in this thesis predicts the dynamics of the OHC from guinea pig since a comprehensive... Physiology of the Ear Summary The anatomy and physiology of the mammalian ear is presented, with an emphasis on the cochlea and organ of Corti The cochlear mechanics is also given, including the traveling waves in the cochlear macro-mechanics, and transduction processes in the cochlear micro-mechanics Finally, the physiology of the outer hair cell is described, together with the electromotile property of the. .. 1: Introduction modes are usually assumed to be unlikely to occur in the in vivo OHC, due to the constraint of the stiff reticular lamina encompassing the OHC cuticular plate at the top The present finding suggests that the in vivo OHC may result in local bending of the reticular lamina by tilting its cuticular plate, producing a dissimilar motion to that in the forward transduction of the organ of. .. brief review of the ear anatomy and functioning of the related components in the ear, with the emphasis on the cochlear mechanics and OHC electromotility Chapter 3 describes the development of the OHC mathematical model using linear composite shell theory, including the coupling between the cell lateral wall and the intra- and extracellular fluids This chapter also presents the determination of length-dependent... cochlea The cochlear micro-mechanics refers to the complicated relative vibration of the elements within the organ of Corti The micro-mechanics involves the functioning of the organ of Corti and underlies the transduction of the mechanical vibration of the basilar membrane into the electrical neural signals in the auditory nerve fibers Moreover, the cochlear amplifier also arises in this stage, providing... malleus, the intermediate incus and the innermost stapes, forming a lever system with the aid of ligaments and muscles in the middle ear The eardrum transmits vibration to a membrane (oval window membrane) of the inner ear, via the malleus, along through the incus to the stapes The middle ear also compensates the impedance mismatch between the sound waves in the external ear and the fluid waves in the inner... traveling wave along the length of the cochlea The tuning property of the basilar membrane results in distinct responses of the membrane, depending on the spectral components of the stimuli and frequency-position map of the cochlea Different 17 Chapter 2: Anatomy and Physiology of the Ear frequencies in sound signals give rise to response peaks occurring at different locations along the length of the cochlea. .. generates the electrical nerve spikes in response to the vibration of the basilar membrane The organ of Corti consists of sensory cells and supporting cells, and its schematic drawing is shown in Figure 2.4 14 Chapter 2: Anatomy and Physiology of the Ear Figure 2.4 Drawing of the anatomy of the organ of Corti (Adapted from Brownell, et al., 2001) Two types of sensory cells, the inner hair cell (IHC) and outer. .. reticular lamina This shearing motion results in the deflection the stereocillia of sensory cells The deflection of the stereocillia opens and closes the mechano-electrical transduction channels An electro-chemical process then arises resulting in the firing of neural signals in the auditory nerve fibers The above process is the forward transduction For the outer hair cell, the deflection of the stereocillia . MATHEMATICAL MODEL OF OUTER HAIR CELLS IN THE COCHLEA LI HAILONG NATIONAL UNIVERSITY OF SINGAPORE 2007 MATHEMATICAL MODEL OF OUTER HAIR CELLS IN THE COCHLEA. knowledge of the dynamics in the cochlea. Finally, a feedback model of the cochlear partition is developed to obtain the OHC activity in the cochlea. Through comparison of the responses in the. the in vitro OHC is first obtained, providing a prerequisite for a better understanding of the dynamics of the in vivo OHC embedded in the cochlea. By including the stiff constraint of the