(L-473) Modeling Creep of Vaginal Tissue in the Longitudinal and Circumferential Directions

Introduction:: Vaginal delivery is a mechanically intensive process which subjects the vagina to large stresses and strains. It is well known that women who deliver vaginally are at high risk of vaginal tearing [1], and they experience higher incidence of pelvic floor disorders than women who are nulliparous or undergo Cesarean delivery [2]. To increase the safety of vaginal delivery, new knowledge is needed about the mechanical response of vaginal tissue to large stresses and strains over long time periods such as those that characterize labor. Our lab has previously shown that vaginal tissue exhibits viscoelastic properties by conducting relaxation and creep tests [3,4]. While experimental studies on creep (i.e., deformation over time at constant load) of vaginal tissue have been performed, no constitutive models have been formulated to describe this experimentally observed viscoelastic behavior. Such models are necessary to fully capture the mechanical response of the vaginal canal to extended loading histories, such as those that are typically observed during prolonged labor. Recently, we subjected rat vaginal canals to three consecutive creep tests via free-extension inflation to different and increasing intraluminal pressures, measuring strains along the longitudinal direction (LD) and circumferential direction (CD) [3]. Here we propose the use of the Pipkin-Rogers viscoelastic framework to mathematically model the results of our biaxial creep experiments [5].


Materials and Methods:: Full details on mechanical testing and results can be found in [3]. Briefly, vaginal canals were dissected from n=14 virgin rats and mounted to a custom-built free-extension inflation set-up. After preconditioning, specimens were inflated to intraluminal pressures of 28, 55, then 83 kPa sequentially, each pressure held for 3000 s. Strains along the LD and the CD were measured throughout testing using 3D DIC software. The tissue exhibited nonlinear mechanical behavior in response to increased loading. In both directions, the tissue experienced greater changes in strain during the first creep test than the second and third creep tests. Notably, the primary creep rate (first 500 s) decreased from the first creep test to later creep tests, while the secondary creep rate (last 2000 s) increased from test to test [3].

Using the dual formulation of Pipkin-Roger (deformation in terms of stress history), we developed a nonlinear-viscoelastic constitutive model that describes the anisotropic creep behavior of vaginal tissue. The model utilizes nonlinear constitutive functions and parameters to capture the stiffening behavior of vaginal tissue in the LD and CD dependent on the current applied pressure. With additional parameters, we can describe the initial compliance of vaginal tissue in the two anatomical directions, as well as the total increase in strain and strain rate during creep. The model parameters were estimated using nonlinear least squares regression to reproduce the entire response of each tested specimen to the full experimental protocol.


Results, Conclusions, and Discussions:: The general modeling framework accounts for the nonlinear elasticity and viscoelasticity, anisotropy, and large deformations of vaginal tissue. For simplicity, we only present the final equation that relate the stretch in the LD, C_zz, as a function of time t and stress history in the LD P_zz(t):

C_zz(t) = 1 + b1·ln(b2·P_zz(t)+1) + g1·∫₀^tln(b2·P_zz(τ)+1)·P_zz(τ)·e^-g2·P_zz^(τ)(t-τ)dτ + a1·∫₀^tP_zz(τ)dτ


where b1 and b2 are LD-specific elastic parameters, g1 and g2 are LD-specific viscoelastic parameters which define the primary creep rate, and a1 is a shared viscoelastic parameter which describes the steady-state secondary creep rate. The equation in the CD is similar, with different direction-specific parameters and dependent on CD stress history.

Figure 1a depicts the experimental results and model fit for one representative specimen in both the LD (blue) and CD (orange). The model provided great fits of the data in both the LD and CD, with a mean adjusted R squared value of 0.9979 across all specimens. Figures 1b, 1c, and 1d depicts the change in stretch versus time during the during the first, second, and third creep tests respectively for the representative specimen. The model does a good job of accurately capturing the transition from early exponential to long-term steady-state linear behavior across all three creep tests.

This model was designed to capture the creep behavior of virgin tissue and may not be applicable to describe the creep behavior of pregnant tissue, as pregnancy induces large microstructural and biomechanical alterations in the vagina [6]. No mechanical studies on viscoelastic behavior have been performed on pregnant vaginal tissue so far, and so it is unknown what changes may be necessary for this model to be able to describe the creep behavior of pregnant tissue.

This work presents the first constitutive model for describing time-dependent deformations of vaginal tissue under constant loading. Furthermore, this marks the first time a constitutive model has been used to describe the response of vaginal tissue to biaxial loading. Future studies, which combine mechanical testing with microstructural imaging techniques, may identify the mechanisms that govern creep behavior in vaginal tissue, guiding future modeling efforts.


Acknowledgements (Optional): : This work was supported by funds from the National Science Foundation (grant numbers 1929731 & 2053851). Rats for testing were provided by the Jarome Lab at Virginia Tech


References (Optional): : [1] E. Samuelsson, L. Ladfors, B. G. Lindblom, H. Hagberg, A prospective observational study on tears during vaginal delivery: occurrences and risk factors, Acta Obstericia et Gynecologica Scandinavica, 81(1):44-49, 2002

[2] Nygaard, I., Shaw, J., Bardsley, T., Egger, M., Lifetime physical activity and pelvic organ prolapse in middle-aged women, American Journal of Obstetrics and Gynecology, 477-e1, 2014

[3] Dubik, J., Tartaglione, A., Miller, K., Dillard, D., De Vita, R., History-dependent deformations of rat vaginas under inflation, Integrative and Comparative Biology, 62(3):625-640, 2022

[4] Pack, E., Dubik, J., Snyder, W., Simon, A., Clark, S., and De Vita, R., Biaxial stress relaxation of vaginal tissue in pubertal gilts, Journal of Biomechanical Engineering, 142(3):031002, 2020

[5] Pipkin, A., Rogers, T., A non-linear integral representation for viscoelastic behaviour, Journal of the Mechanics and Physics of Solids, 16(1):59-72, 1968

[6] Downing, K., Billah, M., Raparia, E., Shah, A., Silverstein, M., Ahmad, A., Boutis, G., The role of mode of delivery on elastic fiber architecture and vaginal vault elasticity, Journal of the mechanical behavior of biomedical materials, 29:190-198, 2014