Lung tissue viscoelasticity: a mathematical framework and its molecular basis.

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.