本文應用Biot多孔彈性動態理論以脈衝回波法分析頻域與時域之海綿骨相速度。研究中首先進行Biot多孔彈性材料動態統御方程組之拉普拉斯轉換，再配合脈衝回波量測之邊界條件，及拉普拉斯轉換因子與角頻率的關係(s=iω)，求得海綿骨頻域之相速度解析解與海綿骨表面位移函數。時域之海綿骨相速度分析則利用海綿骨表面位移頻域函數搭配平頂函數窗之濾波函數，於時域中由波傳時間差計算其相速度。另應用無因次化分析，探討各個無因次化參數於快速波與慢速波無因次相速度的影響。
應用海綿骨之相速度理論解，可探討海綿骨受骨骼材料性質之影響。經與超音波實驗數據比較驗證，本文由理論解預估之相速度結果和趨勢與實驗數據完全吻合。另應用脈衝音壓回波位移頻率響應模擬超音波脈衝回波反應所計算之快速波與慢速波相速度也與解析值完全吻合。本文由海綿骨材料參數變異於脈衝回波之影響分析方面發現固體密度ρs、材料固體架構體積模數Kb與剪力模數N等影響到快速波之相速度；流體體積模數Kf影響到慢速波之相速度；而孔洞係數φ及結構因子α∞對快速波與慢速波皆產生影響；惟有固體體積模數Ks對相速度之影響較不顯著。
另本文由無因次參數變異分析也發現，在無因次快速波相速度部份，無因次彈性係數R*在低頻區影響最大，而P*在高頻區影響最大；此外在無因次慢速波部份則以R*值影響較大。另一方面無因次固體有效密度主要影響快速波相速度；而增加液體有效密度對中頻以後之快速波相速度卻無影響，但會降低慢速波相速度之起始值；另耦合有效密度 值增高，對應之相速度也隨之增高。至於增高無因次消散係數 則會使相速度曲線往高頻移動但不改變曲線形狀。

In this study, Biot’s poroelastic theory is used to derive the pulse echo responses and phase velocities of fast and slow waves of cancellous bones.  First, Biot’s equations were transformed to Laplace domain.  By specifying the boundary conditions for the pulse echo simulation and the relation between the Laplace transform parameter and the angular frequency(s=iω), the theoretical frequency response functions of phase velocities and the displacement of the driving surface of the cancellous bone were obtained.  In time domain, phase velocities of cancellous bone were analyzed using the displacement frequency response function obtained and flat top windows, and were calculated from the rate at which the phase of the wave propagates.  In the end of this study, the influences of dimensionless material parameters on the dimensionless phase velocities of fast and slow waves were discussed.
Using the phase velocity results derived the influences of bone properties on the phase velocities were numerically analyzed.  The predicted results were validated by ultrasonic experimental results and a good agreement was observed.  The phase velocities, which were calculated from the phase propagation rate, also agreed well with the results derived.  After the influence analyses of bone properties on the phase velocities, it was found that the density the solid, the bulk modulus of the frame, the shear modulus of the frame affect the phase velocity of the fast wave.  The bulk modulus the fluid affect the phase velocity of the slow wave.  The porosity and tortuosity of the bone affect both waves.  The only one parameter that has minor effects is the bulk modulus of the solid.
After the analyses of the dimensionless bone properties on the phase velocities, it was found that the dimensionless Biot’s coefficients R* and P* have major effects on the phase velocity of the fast wave in low frequency and high frequency regions, respectively.  Moreover, the dimensionless Biot’s coefficient R* affect the phase velocity of the slow wave.  Furthermore, the dimensionless total effective mass density of the solid affects the phase velocity of the fast wave.  The increase of the dimensionless total effective mass density of the fluid  has no effect on the phase velocity of the fast wave above the low frequency region, but will decrease the initial value of the phase velocity of the slow wave.  Increasing the value of the coupling mass density  will also increase the value of the phase velocities accordingly.  Nevertheless, enlarging the value of the dimensionless dissipation coefficient will move the phase velocity curve toward the high frequency side but makes no change to the shape of the curve.

目  錄
中文摘要	 I
英文摘要	III
目  錄	V
圖目錄	VII
表目錄	X
第一章 緒論	1
1.1 前言	1
1.2 研究動機與目的	2
1.3 文獻回顧	4
1.4 研究內容	7
第二章 多孔彈性力學與材料參數	8
2.1 Biot多孔彈性理論	8
2.1.1 應力、應變及位移	9
2.1.2 Biot彈性係數	10
2.2 多孔材料參數	12
2.2.1 孔洞係數	12
2.2.2 彈性係數與材料係數之關係	12
2.2.3 結構因子	15
2.2.4 動態消散係數	16
2.2.5 空氣之體積模數	16
第三章 海綿骨之相速度分析	18
3.1 海綿骨之位移及應力函數	18
3.1.1 頻域海綿骨相速度解析解	19
3.1.2 海綿骨表面位移頻域函數	20
3.1.3 海綿骨相速度頻域分析	24
3.1.4海綿骨材料性質於其相速度之影響分析	28
3.2 海綿骨之時域脈衝回波模擬與相速度量測	31
3.2.1 函數窗	31
3.2.2時域脈衝回波模擬與相速度量測	33
3.2.3海綿骨材料參數變異於脈衝回波之影響	37
第四章 海綿骨相速度無因次化分析	50
4.1 無因次相速度	50
4.2 無因次化參數變異之影響分析	53
4.2.1 無因次彈性係數變異於快速波無因次相速度之影響	53
4.2.2 無因次彈性係數變異於慢速波無因次相速度之影響	55
4.2.3 無因次有效密度變異於快速波與慢速波無因次相速度之影響	57
第五章 結論與未來展望	65
5.1 結論	65
5.2 未來展望	67
參考文獻	68
符號定義對照表	73
 
圖目錄
圖1-1：穿透傳送法示意圖	3
圖1-2：脈衝回波法示意圖	4
圖2-1：傳導波入射於多孔彈性材料反應示意圖	9
圖2-2：(a)α∞近似於1，(b)α∞大於1	15
圖3-1：脈衝音波入射貼於剛性壁上之海綿骨示意圖	23
圖3-2：海綿骨5之相速度頻域解析解與實驗結果比較	27
圖3-3：不同流體於海綿骨相速度之影響	29
圖3-4：不同骨質於海綿骨相速度之影響	29
圖3-5：不同孔洞係數海綿骨之相速度	30
圖3-6：函數窗w(t)濾波示意圖	31
圖3-7：中心頻率2 MHz平頂函數窗	32
圖3-8：中心頻率2 MHz平頂函數窗(dB值)	32
圖3-9：1.0 MHz超音波脈衝回波模擬結果(海綿骨5)	34
圖3-10：1.5 MHz超音波脈衝回波模擬結果(海綿骨5)	34
圖3-11：2.0 MHz超音波脈衝回波模擬結果(海綿骨5)	35
圖3-12：2.5 MHz超音波脈衝回波模擬結果(海綿骨5)	35
圖3-13：3.0 MHz超音波脈衝回波模擬結果(海綿骨5)	36
圖3-14：2 MHz超音波反射式量測回波模擬(海綿骨6)	38
圖3-15：孔洞係數 變異於超音波脈衝回波之影響	39
圖3-16：孔洞係數 變異於超音波脈衝回波之影響(局部放大)	39
圖3-17：結構因子 變異於超音波脈衝回波之影響	41
圖3-18：結構因子 變異於超音波脈衝回波之影響(局部放大)	41
圖3-19：固體密度 變異於超音波脈衝回波之影響	42
圖3-20：固體密度 變異於超音波脈衝回波之影響(局部放大)	42
圖3-21：流體體積模數 變異於超音波脈衝回波之影響	44
圖3-22：流體體積模數 變異於超音波脈衝回波之影響(局部放大)	44
圖3-23：固體體積模數 變異於超音波脈衝回波之影響	45
圖3-24：固體體積模數 變異於超音波脈衝回波之影響(局部放大)	45
圖3-25：固體架構體積模數 變異於超音波脈衝回波之影響	47
圖3-26：固體架構體積模數 變異於超音波脈衝回波之影響(局部放大)	47
圖3-27：剪力模數 變異於超音波脈衝回波之影響	48
圖3-28：剪力模數 變異於超音波脈衝回波之影響(局部放大)	48
圖4-1： 於快速波無因次相速度之影響	54
圖4-2： 於快速波無因次相速度之影響	54
圖4-3： 於快速波無因次相速度之影響	55
圖4-4： 於慢速波無因次相速度之影響	56
圖4-5： 於慢速波無因次相速度之影響	56
圖4-6： 於慢速波無因次相速度之影響	57
圖4-7： 於快速波無因次相速度之影響	58
圖4-8： 於快速波無因次相速度之影響(局部放大)	58
圖4-9： 於慢速波無因次相速度之影響	59
圖4-10： 於快速波無因次相速度之影響	60
圖4-11： 於慢速波無因次相速度之影響	60
圖4-12： 於慢速波無因次相速度之影響(局部放大)	61
圖4-13： 於快速波無因次相速度之影響	62
圖4-14： 於快速波無因次相速度之影響(局部放大)	62
圖4-15： 於慢速波無因次相速度之影響	63
圖4-16： 於快速波無因次相速度之影響	64
圖4-17： 於慢速波無因次相速度之影響	64
 
表目錄
表3-1：海綿骨骨架材料表	26
表3-2：1~3 MHz海綿骨快速波之相速度解析解(m/s)	27
表3-3：1~3 MHz海綿骨慢速波之相速度解析解(m/s)	27
表3-4：海綿骨之流體材料性質	28
表3-5：海綿骨5之相速度模擬結果	36
表3-6：海綿骨6及其20%參數變異值	37
表4-1：海綿骨1~8之無因次值	52
表4-2：無因次分析列表	53

參考文獻
1.D. Hans, P. Dargent-Molina, A. M. Schott and et al., “Ultrasonographic Heel Measurements to Predict Hip Fracture in Elderly Women: The EPIDOS Prospective Study”, The Lancet, Vol. 348, pp. 511-514, 1996.
2.S. Grampp, C. Henk, Y. Lu and et al., “Quantitative US of the Calcaneus: Cutoff Levels for the Distinction of Healthy and Osteoporotic Individuals”, Radiology, Vol. 220, No. 2, pp. 400-405, 2001.
3.N. Sebaa, Z. E. A. Fellah, W. Lauriks and C. Depollier, “Application of Fractional Calculus to Ultrasonic Wave Propagation in Human Cancellous Bone”, Signal Processing, Vol. 86, No. 10, pp. 2668-2677, 2006.
4.A. Hosokawa and T. Otani, “Ultrasonic Wave Propagation in Bovine Cancellous Bone”, The Journal of the Acoustical Society of America, Vol. 101, No. 1, pp. 558-562, 1997.
5.Z. E. A. Fellah, J. Y. Chapelon, S. Berger, W. Lauriks and C. Depollier, “Ultrasonic Wave Propagation in Human Cancellous Bone: Application of Biot Theory”, The Journal of the Acoustical Society of America, Vol. 116, No. 1, pp. 61-73, 2004.
6.E. R. Hughes, T. G. Leighton, G. W. Petley and P. R. White, “Ultrasonic Propagation in Cancellous Bone: A New Stratified Model”, Ultrasound in Medicine & Biology, Vol. 25, No. 5, pp. 811-821, 1999.
7.A. D. Dimarogonas, “Vibration for Engineers”, Prentice-Hall International, 1992.
8.K. Terzaghi, “Theoretical Soil Mechanics”, Wiley, New York, 1943.
9.M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid- Saturated Porous Solid. I. Low-Frequency Range”, The Journal of the Acoustical Society of America, Vol. 28, No. 2, pp. 168-178, 1956.
10.M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. II. Higher Frequency Range”, The Journal of the Acoustical Society of America, Vol. 28, No. 2, pp. 179-191, 1956.
11.M. A. Biot and D. G. Willis, “The Elastic Coefficients of the Theory of Consolidation”, The Journal of Applied Mechanics, Vol. 24, pp. 594-601, 1957.
12.T. J. Plona, “Observation of a Second Bulk Compressional Wave in a Porous Medium at Ultrasonic Frequencies”, Applied Physics Letters Vol. 36, No. 4, pp. 259-261, 1980.
13.J. G. Berryman, “Confirmation of Biot's Theory”, Applied Physics Letters, Vol. 37, No. 4, pp. 382-384, 1980.
14.R. B. Ashman, J. D. Corin and C. H. Turner, “Elastic Properties of Cancellous Bone: Measurement by an Ultrasonic Technique”, Journal of Biomechanics, Vol. 20, No. 10, pp. 979-986, 1987.
15.R. B. Ashman and J. Y. Rho, “Elastic Modulus of Trabecular Bone Material”, Journal of Biomechanics, Vol. 21, No. 3, pp. 177-181, 1988.
16.S. B. Lan, “Elastic Coefficients of Animal Bone”, Science, Vol. 165, pp. 287-288, 1969.
17.W. C. Van Buskirk, S. C. Cowin and R. N. Ward, “Ultrasonic. Measurement of Orthotropic Elastic Constants of Bovine Femoral Bone”, Journal of Biomechanical Engineering, Vol. 103, No. 2, pp. 67-71, 1981.
18.L. J. Gibson, “The Mechanical Behaviour of Cancellous Bone”, Journal of Biomechanics, Vol. 18, No. 5, pp. 317-328, 1985.
19.A. Hosokawa and T. Otani, “Acoustic Anisotropy in Bovine Cancellous Bone”, The Journal of the Acoustical Society of America, Vol. 103, No. 5, pp. 2718-2722, 1998.
20.K. Mizuno, M. Matsukawa, T. Otani, M. Takada, I. Mano and T. Tsujimoto, “An Experimental Study on the Ultrasonic Wave Propagation and Structural Anisotropy in Bovine Cancellous Bone” Proceedings-IEEE Ultrasonics Symposium, pp. 2163-2166, 2007.
21.P. W. Thomson, J. Taylor, R. Oliver and A. Fisher, “Quantitative Ultrasound(QUS) of the Heel Predicts Wrist and Osteoporosis- Relatedf Ractures in Women Age 45-75 Years”, Journal of Clinical Densitometry, Vol. 1, No. 3, pp. 219-225, 1998.
22.D. C. Bauer, C. C.  , J. A. Cauley, T. M. Vogt, K. E. Ensrud, H. K. Genant and D. M. Black, “Broadband Ultrasound Attenuation Predicts Fractures Strongly and Independently of Densitometry in Order Women”, Archives of Internal Medicine, Vol. 157, No. 6, pp. 629-634, 1997.
23.T. J. McKelvie and S. B. Palmer, “The Interaction of Ultrasound with Cancellous Bone”, Physics in Medicine and Biology, Vol. 36, No. 1, pp. 1331-1340, 1991.
24.J. L. Williams “Ultrasonic Wave Propagation in Cancellous and Cortical Bone Prediction of Some Experimental Results by Biot's Theory”, The Journal of the Acoustical Society of America, Vol. 91, No. 2, pp. 1106-1112, 1992.
25.M. A. Biot, “General Theory of Three Dimensional Consolidation”, Journal of Applied Physics, Vol. 12, No. 2, pp. 155-164, 1941.
26.H. S. Tsay and H. F. Yeh, “Frequency Response Function for Prediction of Planar Cellular Plastic Foam Acoustic Behavior”, Journal of Cellular Plastics, Vol. 41, No. 2, pp. 101-131, 2005.
27.H. S. Tsay, “Dynamics Response to Impulsive Loads of Structures Composed of Poroelastic Materials”, PhD Thesis, University of Delaware, 1990.
28.J. F. Allard, “Propagation of Sound in Porous Media Modeling: Sound Absorbing Material”, Elsevier Science Publisher, England., 1993.
29.C. Zwikker and C. W. Kosten, “Sound Absorbing Materials”, Elasevier, 1949.
30.H. Lamb, “Hydordynamics ”, Dover, New York, pp. 152-156, 1945.
31.A. Craggs and J. G. Hildebrandt, “Effective Densities and Resistivities for Acoustic Propagation in Narrow Tube”, Journal of Sound and Vibration, Vol. 92, No. 3, pp. 321-331, 1984.
32.A. Craggs and J. G. Hildebrandt, “The Normal Incidence Absorption Coefficient of a Matrix of Narrow Tubes with Constant Cross- Section”, Journal of Sound and Vibration, Vol. 105, No. 1, pp. 641-652, 1986.
33.Y. Champoux and J. F. Allard, “Dynamic Tortuosity and Bulk Modulus in Air-Satured Porous Media”, Journal of Applied Physics, Vol. 70, No. 4, pp. 1975-1979, 1991.
34.M. R. Stinson, “The Propagation of Plane Sound Waves in Narrow and Wide Circular Tubes, and Generalization to Uniform Tubes of Arbitrary Cross-Sectional Shape”, The Journal of the Acoustical Society of America, Vol. 89, No. 2, pp. 550-558, 1991.
35.K. I. Lee and S. W. Yoon, “Modifications of the Biot and the MBA Models for Predicting the Anisotropic Phase Velocity in Cancellous Bone”, Journal of the Korean Physical Society, Vol. 48, No. 4, pp. 564-572, 2006.
36.K. I. Lee and S. W. Yoon, “Comparison of Acoustic Characteristics Predicted by Biot's Theory and the Modified Biot-Attenborough Model in Cancellous Bone”, Journal of Biomechanics, Vol. 39, No. 2, pp. 364-368, 2006.
37.A. Hosokawa, “Simulation of Ultrasound Propagation Through Bovine Cancellous Bone Using Elastic and Biot’s Finite-Difference Time-Domain Methods”, The Journal of the Acoustical Society of America, Vol. 118, No. 3, pp. 1782-1789, 2005.
38.K. I. Lee, H. S. Roh, B. N. Kim, and S. W. Yoon, “Acoustic Characteristics of Cancellous Bone: Application of Biot's Theory and the Modified Biot-Attenborough Model”, Journal of the Korean Physical Society, Vol. 45, No. 2, pp. 386-392, 2004.
39.K. R. Marutyan, M. R. Holland, and J. G. Miller, “Anomalous Negative Dispersion in Bone Can Result from the Interference of Fast and Slow Waves”, The Journal of the Acoustical Society of America, Vol. 120, No. 5, pp. EL55-EL61, 2006.
40.D. L. Johnson, J. Koplik and R. Dashen, “Theory of Dynamic Permeability and Tortuosity in Fluid-Saturated Porous Media”, Journal of Fluid Mechanics, Vol. 176, pp. 379-402. 1987.
41.Z. E. A. Fellah, M. Fellah, W. Lauriks, and C. Depollier, “Direct and Inverse Scattering of Transient Acoustic Waves by a Slab of Rigid Porous Material”, The Journal of the Acoustical Society of America, Vol. 113, No. 1, pp. 61-72, 2003.