Advancing Maternal Health through Projection-based and Machine Learning Strategies for Reduced Order Modeling

High-fidelity computer simulations of childbirth are time consuming, making them impractical for guiding decision-making during obstetric emergencies. The complex geometry, micro-structure, and large finite deformations undergone by the vagina during childbirth result in material and geometric nonlinearities, complicated boundary conditions, and nonhomogeneities within finite element (FE) simulations. Such nonlinearities pose a significant challenge for numerical solvers, increasing the computational time. Simplifying assumptions can reduce the computational time significantly, but this usually comes at the expense of simulation accuracy. The work herein proposed the use of reduced order modeling (ROM) techniques to create surrogate models that capture experimentally-measured displacement fields of rat vaginal tissue during inflation testing in order to attain both the accuracy of higher-fidelity models and the speed of lower-fidelity simulations. The proper orthogonal decomposition (POD) method was used to extract the significant information from FE simulations generated by varying the luminal pressure and the parameters that introduce the anisotropy in the selected constitutive model. In our first study, a new data-driven (DD) variational multiscale (VMS) ROM framework was extended to obtain the displacement fields of rat vaginal tissue subjected to ramping luminal pressure. For comparison purposes, we also investigated the classical Galerkin ROM (G-ROM). In our numerical study, both the G-ROM and the DD-VMS-ROM decreased the FE computational cost by orders of magnitude without a significant decrease in numerical accuracy. Furthermore, the DD-VMS-ROM improved the G-ROM accuracy at a modest computational overhead. Our numerical investigation showed that ROM had the potential to provide efficient and accurate computational tools to describe vaginal deformations, with the ultimate goal of improving maternal health. Our second study compared two common computational strategies for surrogate modeling, physics-based G-ROM and data-driven machine learning (ML), for decreasing the cost of FE simulations of the ex vivo deformations of rat vaginal tissue subjected to inflation testing to study the effect of a pre-imposed tear. Since there are many methods associated with each modeling approach, to provide a fair and natural comparison, we selected a basic model from each category. From the ROM strategies, we considered a simplified G-ROM that is based on the linearization of the underlying nonlinear FE equations. From the ML strategies, we selected a feed-forward dense neural network (DNN) to create mappings from constitutive model parameters and luminal pressure values to either the FE displacement history (in which case we denote the resulting model ML) or the POD coefficients of the displacement history (in which case we denote the resulting model POD-ML). The numerical comparisons of G-ROM, ML, and POD-ML took place in the reconstructive regime. The numerical results showed that the G-ROM outperformed the ML model in terms of offline central processing unit (CPU) time for model training, online CPU time required to generate approximations, and relative error with respect to the FE models. The POD-ML model improved on the speed performance of the ML, having online CPU times comparable to those of the G-ROM given the same size of POD bases. However, the POD-ML model did not improve on the error performance of the ML. In our last study, we expanded our investigation of ML methods for surrogate modeling by comparing the performance of a DNN similar to what was used previously to that of a convolutional neural network (CNN) using 1-D convolution on the input parameters from FE simulations of active vaginal tearing. The new FE simulations utilized a custom continuum damage model that provided material damage and failure properties to an existing anisotropic hyperelastic constitutive model to replicate experimentally-observed tear propagation behaviors. We employed our DNN and CNN models to create mappings from constitutive model parameters, geometric properties of the propagating tear, and luminal pressure values to either the full FE displacement history or the POD coefficients of the displacement history. The root-mean-square error (RMSE) with respect to the FE displacement history achieved by full order output ML predictions was reproducible with POD-ML using a basis of only dimension l=10. Additionally, an order of magnitude reduction in offline time was observed using POD-ML over full-order ML with minimal difference between DNN and CNN architectures. Differences in online computational costs between ML and POD-ML were found to be negligible, but the DNNs produced predictions slightly faster than the CNNs, though both online times were on the same order of magnitude. While convolution did not significantly aid the regression task at hand, POD-ML was demonstrated to be an efficient and effective approach for surrogate modeling of the FE tear propagation model, approximating the displacement history with RMSE less than 0.1 mm and generating results 7 orders of magnitude faster than the FE model. This set of baseline numerical investigations serves as a starting point for future computer simulations that consider state-of-the-art G-ROM and ML strategies, and the in vivo geometry, boundary conditions, material properties, and tissue damage mechanics of the human vagina, as well as their changes during labor.