Lung tissue viscoelasticity: a mathematical framework and its molecular basis

Area of Science:

• Biophysics
• Materials Science
• Rheology

Background:

• Lung tissue stress relaxation and impedance are empirically described by power law and constant-phase models, outperforming traditional spring-dashpot systems.
• These empirical models lack a clear mechanistic basis, limiting deeper understanding of soft tissue viscoelasticity.

Purpose of the Study:

• To develop a mathematical framework for power law relaxation and constant-phase impedance in soft tissues.
• To explore the mechanistic basis of these phenomena using fractional calculus and polymer physics.
• To relate key parameters to molecular-level dynamics in soft tissues.

Main Methods:

• Utilized fractional calculus by replacing ordinary time derivatives with fractional time derivatives in constitutive equations.
• Analyzed the Fourier transform of the resulting fractional relaxation function to derive the constant-phase impedance.
• Investigated molecular theories of polymer systems to establish a mechanistic link to fractional derivatives and tissue viscoelasticity.

Main Results:

• Fractional time derivatives naturally yield power law relaxation functions and constant-phase impedance, with a direct relationship between exponents (alpha = 1 - beta).
• Established a mechanistic basis for fractional derivatives in polymer viscoelasticity, stemming from molecular complexity and statistical properties.
• The exponent beta is directly linked to dynamic processes at the tissue fiber and matrix level.

Conclusions:

• Fractional calculus offers a robust mathematical framework for describing soft tissue viscoelasticity, unifying empirical observations with mechanistic insights.
• The molecular theories of polymer systems provide a potential explanation for the observed mechanical properties of soft tissues.
• This approach deepens the understanding of soft tissue mechanics by connecting macroscopic behavior to molecular-level dynamics.