Suki B, Barabási A L, Lutchen K R
Department of Biomedical Engineering, Boston University, Massachusetts 02215.
J Appl Physiol (1985). 1994 Jun;76(6):2749-59. doi: 10.1152/jappl.1994.76.6.2749.
Recent studies indicated that lung tissue stress relaxation is well represented by a simple empirical equation involving a power law, t-beta (where t is time). Likewise, tissue impedance is well described by a model having a frequency-independent (constant) phase with impedance proportional to omega-alpha (where omega is angular frequency and alpha is a constant). These models provide superior descriptions over conventional spring-dashpot systems. Here we offer a mathematical framework and explore its mechanistic basis for using the power law relaxation function and constant-phase impedance. We show that replacing ordinary time derivatives with fractional time derivatives in the constitutive equation of conventional spring-dashpot systems naturally leads to power law relaxation function, the Fourier transform of which is the constant-phase impedance with alpha = 1 - beta. We further establish that fractional derivatives have a mechanistic basis with respect to the viscoelasticity of certain polymer systems. This mechanistic basis arises from molecular theories that take into account the complexity and statistical nature of the system at the molecular level. Moreover, because tissues are composed of long flexible biopolymers, we argue that these molecular theories may also apply for soft tissues. In our approach a key parameter is the exponent beta, which is shown to be directly related to dynamic processes at the tissue fiber and matrix level. By exploring statistical properties of various polymer systems, we offer a molecular basis for several salient features of the dynamic passive mechanical properties of soft tissues.
最近的研究表明，肺组织的应力松弛可以用一个涉及幂律t-β（其中t是时间）的简单经验方程很好地表示。同样，组织阻抗可以用一个模型很好地描述，该模型具有与频率无关（恒定）的相位，且阻抗与ω-α成正比（其中ω是角频率，α是一个常数）。这些模型比传统的弹簧-阻尼系统提供了更好的描述。在这里，我们提供一个数学框架，并探索其使用幂律松弛函数和恒定相位阻抗的机制基础。我们表明，在传统弹簧-阻尼系统的本构方程中用分数阶时间导数代替普通时间导数自然会导致幂律松弛函数，其傅里叶变换是α = 1 - β的恒定相位阻抗。我们进一步确定分数阶导数对于某些聚合物系统的粘弹性具有机制基础。这种机制基础源于在分子水平上考虑系统复杂性和统计性质的分子理论。此外，由于组织由长的柔性生物聚合物组成，我们认为这些分子理论也可能适用于软组织。在我们的方法中，一个关键参数是指数β，它被证明与组织纤维和基质水平的动态过程直接相关。通过探索各种聚合物系统的统计性质，我们为软组织动态被动力学性质的几个显著特征提供了分子基础。
