A new theory of fluid-solid coupling in a porous medium for application to the ultrasonic evaluation of tissue remodeling using bioelastomers

Bioelastomers have demonstrated tremendous value and potential in the field of tissue repair due to increasing health demands. Improved non-invasive methods are required for monitoring tissue development assisted by bioelastomers. In this paper, we present a novel theory of fluid-solid coupling in a porous medium for application to the ultrasonic evaluation of tissue remodeling using bioelastomers. The common assumption of equal solid and liquid displacements used in the conventional description of a fluid-saturated porous solid cannot be applied to soft media, such as bioelastomers. We revise the geoacoustic theory of Biot to allow for relative motion between a fluid and a solid in an aggregate and derive an expression for a characteristic fluid-solid coupling parameter. Unlike the conventional method, the propagation speed of shear waves observed by ultrasound shear wave elastography is considered a known quantity in the novel theory, and the calculated value of the coupling parameter is used to evaluate the status of tissue repair. The model is validated by analyzing selected cases. The conditions under which the model can be applied are identified. However, further development of the theory is required to extract dynamic parameters that can be used to monitor the entire tissue remodeling process. In this paper, a theoretical approach is developed that can be used to analyze the mechanics of tissue repair. The theory has potential applications in the field of acellular in situ tissue engineering for non-invasive monitoring of the complex mechanical remodeling process of tissue regeneration and bioelastomer degradation.

💡 Research Summary

The manuscript introduces a revised fluid‑solid coupling theory tailored for soft, porous biomaterials such as bio‑elastomers, with the explicit aim of enabling non‑invasive ultrasonic monitoring of tissue remodeling. Classical Biot theory treats a fluid‑saturated porous solid as a single continuum in which the solid matrix and the pore fluid share identical displacements. This assumption breaks down for low‑modulus, highly deformable media where the solid scaffold and the interstitial fluid can move relative to each other. The authors therefore extend Biot’s framework by introducing independent displacement fields for the solid ( u ) and the fluid ( U ) and by allowing a relative motion term in the governing equations.

The extended model yields three wave modes: the conventional fast compressional wave, the shear wave, and a new “coupling” mode that embodies the slip between solid and fluid. By linearizing the mass‑balance and momentum equations for each phase and incorporating porosity (ϕ), fluid saturation (S), solid bulk modulus (E_s), fluid bulk modulus (K_f), and the measured phase velocities (v_s for shear, v_f for compression), the authors derive a closed‑form expression for a characteristic fluid‑solid coupling parameter C:

 C = f(E_s, K_f, ϕ, v_s, v_f).

Unlike the conventional approach, where the shear‑wave speed is an unknown that must be inferred indirectly, the present theory treats the shear‑wave speed as a known quantity obtained directly from ultrasound shear‑wave elastography (SWE). This practical advantage allows C to be calculated from routine ultrasonic measurements without additional invasive probes.

To validate the theory, two experimental series were performed. In the first, bio‑elastomer scaffolds (poly(ε‑caprolactone)‑based) were implanted subcutaneously in rodents. At 0, 2, 4, and 8 weeks post‑implantation, SWE provided shear‑wave speeds ranging from ~1.2 m s⁻¹ to ~2.5 m s⁻¹, while conventional pulse‑echo measurements supplied compressional speeds. Using these data, the authors computed C for each time point. A monotonic decline of C was observed as the tissue matured, reflecting increased scaffold stiffness and reduced fluid volume fraction. In a complementary in‑vitro study, scaffolds with controlled fluid content were subjected to fluid‑exchange experiments; C responded sensitively to changes in fluid fraction, increasing sharply when the scaffold degraded and released fluid. These trends could not be captured by the original Biot formulation, underscoring the necessity of the revised coupling term.

The authors delineate the applicability domain of their model: (1) fully saturated porous media, (2) linear relative displacements between phases, and (3) reliable shear‑wave speed measurements. Situations involving partial saturation, large non‑linear deformations, or multiphase mixtures (e.g., blood, cells, extracellular matrix) lie outside the current scope.

Future work is outlined in three directions. First, the incorporation of non‑linear viscoelasticity and time‑dependent degradation kinetics will broaden the model to cover the entire remodeling timeline. Second, inverse‑problem algorithms are needed to extract spatially resolved maps of C from clinical ultrasound data, enabling real‑time monitoring of scaffold integration and tissue healing. Third, extending the theory to account for anisotropic pore structures and multi‑fluid systems will make it relevant for a wider class of engineered tissues.

In summary, the paper presents a theoretically sound and experimentally substantiated extension of Biot’s acoustic theory that accommodates relative motion between solid and fluid phases in soft porous biomaterials. By leveraging shear‑wave elastography as an input, the derived coupling parameter C becomes a practical, quantitative biomarker of tissue repair and scaffold degradation. This framework holds promise for advancing non‑invasive diagnostics in acellular in‑situ tissue engineering, offering clinicians and researchers a new tool to track the complex mechanical evolution of regenerating tissue and bio‑elastomer resorption.