使用有限元素分析模擬肌腱的應變硬化性質與彈性影像的結果之比較

Abstract

肌腱的功能是將肌肉產生的力量傳遞到骨頭以協助肢體的移動，並儲存或釋放來自肌肉產生的能量以助調節力量，使肌肉有較良好的力量輸出而不致使受傷。肌腱功能的正常性仰賴肌腱合適的拉伸彈性，也就是肌腱拉伸方向的楊氏係數Ｅ，肌腱的傷害來自於過度的使用並且與活動時肌腱所承受的力量息息相關，因此研究肌腱承受不同力量時肌腱的彈性變化，有助於臨床上分辨正常與病變肌腱並追蹤治療成效。近年來，使用超音波彈性影像定量組織內彈性已經被廣泛應用，尤其在等向性、均質性和線性彈性組織特性下，可以從剪力模數推導出楊氏係數．但是在我們先前單軸拉伸實驗和彈性影像量測中發現，肌腱拉伸彈性與沿著拉伸方向傳遞的剪切波波速兩者皆會隨著施加其上的拉力增大(0—3N)而近似線性增加 ，正常肌腱的彈性從2.69 MPpa增加到13.78 MPpa，剪切波速度從7.29 m/s上升到21.40 m/s，而病變肌腱的彈性從1.43 MpPa增加到8.5 MPpa，剪切波速度從6.01 m/s上升到17.74 m/s．但是透過單軸拉伸所得的彈性無法和一般組織之剪切波波速轉換彈性公式做關聯(也就是彈性等於三倍密度乘上波速平方，E=3ρ(v_s)^2．主要原因來自於肌腱存在非等向性結構和非線性彈性特性。此外，若引入橫向等向性模型來定量肌腱彈性時，則需量測五個參數：C_11, C_13, C_33, C_44和 C_66 ，這些參數可透過超音波縱橫波波速量測而得 ．然而，實際上波速的量測過於困難。為了更精確描述肌腱彈性並輔助建立彈性影像，本文使用橫向等向性(transverse isotropy)模型描述其纖維方向及等向性平面，並利用超彈性(hyperelastic)模型定義其非線性應力應變曲線，透過ABAQUS來模擬肌腱單軸拉伸時，肌腱彈性的動態變化以及各方向剪切波波速的改變。模擬結果顯示當肌腱承受0—3 N沿長軸方向拉力時，正常肌腱模擬波速從15.9 m/s增加至23.61 m/s，病變肌腱模擬波速則從14.26 m/s增加到16.43 m/s，亦即模擬剪切波波速隨施加拉力變化的趨勢與離體肌腱實驗結果相同，並且在較高拉力時，模擬波速的數值量級與離體肌腱實驗結果相當接近。這些結果表示適當合併橫向等向性與超彈性之力學模型可用於解釋並評估肌腱在受力下之彈性變化。總言之，我們利用有限元素分析來解釋由剪切波彈性影像的量測到的肌腱應變硬化行為，並應用此模型研究受到不同應力時，縱橫波波速變化，此結果有助於臨床上運用剪切波彈性影像定量評估肌腱傷害並追蹤復原程度。

The function of tendon is to transmit the energy generated by muscles to the bone to help body movement. Tendons play a key role to regulate the force output by releasing or storing the energy in order to present from avoid taking damages. These functionalities depends on its proper tensile stiffness, i.e., the Young's modulus E of the tendon along the stretching direction of the tendon. Tendon injuries areoften result from an over-use disease, which and isare closely related to the mechanical loading imposed on the tendon during physical activity. Therefore, to study the changes of tendons stiffness with respect to various external loads will help us distinguish between normal and pathological state of tendon and track the effectiveness of treatment. In recent years, the ultrasound-based shear wave elasticity imaging has been widely used to quantify tissue elasticity. Especially in the isotropic, homogeneous and linearly elastic medium, the Young’s modulus can be approximately derived from shear modulus. In our previous ex vivo experiments of SWEI and tensile test, we clearly demonstrated both the shear wave velocities and tensile moduli of the normal/injured tendons increased as the pre-stretches increased. When the tendons were preloaded from 0 to 3N, the tensile moduli of the samples increased from 2.69 to 13.78 MPa, while the mean velocities of shear waves propagating along the longitudinal axis of the tendons increased from 7.29 to 21.40 m/s, Likewise, the tensile moduli in the injured tendons increased from 1.43 to 8.5 MPa as the preloads increased, while the mean velocities of shear waves propagating increased from 6.01 to 17.74 m/s. However, the Young’s modulus derived from shear wave velocities (i.e. E=3ρ(v_s)^2) cannot’t be coincided with the Young’s modulus measured by uniaxial tensile test. In other words, there are few quantitative models available to interpret the SWEI results measured in tendons due to their complex architecture and nonlinear mechanical behaviors. In addition, if we use the transverse isotropic model to characterize the mechanical properties of tendon, we need to measure five elastic constants, C_11, C_13, C_33,C_44 and C_66. These constants can be obtained by measuring the longitudinal and shear wave velocities propagating through tendons. However, in practical use, it is difficult to measure these constants. In order to assist us in building elasticity imaging of tendon, in this study, we used the transverse isotropic model and hyperelastic model to describe the fiber orientation and strain-stiffening behaviors of tendons. A transverse isotropic hyperelastic model using ABAQUS was employed and shear wave propagation was simulated in the modeled tendons when they were pre-stretched by loads varying from 0 to 3 N. Our preliminary results successfully recapitulated the trend of changes of shear wave velocities with respect to different pre-stretches observed in SWEI. The simulated velocity of shear waves propagating along the longitudinal axis of the control tendons increased from 15.9 to 23.61 m/s. On the other hand, the simulated velocity of shear waves propagating along the longitudinal axis of the injured tendons increased from 14.26 to 16.43 m/s. In short, the simulated velocities with respect to various loads in this model agreed well with that measured in ex vivo experiments, especially in the higher stressed level. These results show that merging transverse isotropic model and hyperelastic model appropriately can be used to interpret the changes of elasticity when tendons subjected to loads. Our work provides a quantitative basis to explain the strain-stiffening behaviors of tendons measured by SWEI and highlights the potential of applying SWEI to quantitatively assess mechanical dysfunction of injured tendons.