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Abstract We present our continuous efforts from a modeling and numerical viewpoint to develop a powerful and flexible mathematical and computational framework called Ocular Mathematical Virtual Simulator (OMVS). The OMVS aims to solve problems arising in biomechanics and hemodynamics within the human eye. We discuss our contribution towards improving the reliability and reproducibility of computational studies by performing a thorough validation of the numerical predictions against experimental data. The OMVS proved capable of simulating complex multiphysics and multiscale scenarios motivated by the study of glaucoma. Furthermore, its modular design allows the continuous integration of new models and methods as the research moves forward, and supports the utilization of the OMVS as a promising non‐invasive clinical investigation tool for personalized research in ophthalmology. 

This paper proposes a method to learn ap- proximations of missing Ordinary Differential Equations (ODEs) and states in physiological models where knowl- edge of the system’s relevant states and dynamics is in- complete. The proposed method augments known ODEs with neural networks (NN), then trains the hybrid ODE-NN model on a subset of available physiological measurements (i.e., states) to learn the NN parameters that approximate the unknown ODEs. Thus, this method can model an ap- proximation of the original partially specified system sub- ject to the constraints of known biophysics. This method also addresses the challenge of jointly estimating physio- logical states, NN parameters, and unknown initial condi- tions during training using recursive Bayesian estimation. We validate this method using two simulated physiolog- ical systems, where subsets of the ODEs are assumed to be unknown during the training and test processes. The proposed method almost perfectly tracks the ground truth in the case of a single missing ODE and state and performs well in other cases where more ODEs and states are missing. This performance is robust to input signal per- turbations and noisy measurements. A critical advantage of the proposed hybrid methodology over purely data-driven methods is the incorporation of the ODE structure in the model, which allows one to infer unobserved physiological states. The ability to flexibly approximate missing or inac- curate components in ODE models improves a significant modeling bottleneck without sacrificing interpretability. 

In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the surrounding blood circulation. In this paper, we propose a heterogeneous model where a local, accurate, 3D description of tissue perfusion by means of fluid flows through deformable porous media equations is coupled with a systemic, 0D, lumped model of the remainder of the circulation, where the fluid flow through a vascular network is described via its analog with a current flowing through an electric circuit. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific tissue region with an initial value problem in the surrounding circulatory system. This PDE/ODE coupling leads to interface conditions enforcing the continuity of mass and the balance of stresses across models at different scales, and careful consideration is taken to address this interface mismatch. The resulting system involves PDEs of mixed type with interface conditions depending on nonlinear ODEs. A new result on local existence of solutions for this multiscale interface coupling is provided in this article. 

Many pathologies are related to hemodynamic changes occurring at the microvascular level, where small vessels pierce the tissue, perfusing it with blood. Since there is a large number of vessels of small caliber, it is impractical to model the fluid flow through each one of them separately, as it is done in the case of large arteries using, for example, the Navier–Stokes equations. As an alternative, tissue perfusion is modeled here via three‐dimensional (3D) partial differential equations (PDEs) for fluid flow through deformable porous media, where blood vessels are modeled as pores within a deformable solid representing the tissue. Since it is known that the local perfusion is related to the systemic features of surrounding blood circulation, we couple the PDE system with a zero‐dimensional (0D) lumped circuit model, obtained by the analogy between fluid flows in hydraulic networks and current flowing in electrical circuits. An important feature in this multiscale 3D–0D coupling is the specification of interface conditions between the 3D and the 0D parts of the system. In this article, we focus on two types of interface conditions driven by physical considerations, and compare the behavior of the solutions for the two different scenarios. 

The ability of a cell to keep its volume constant irrespective of intra- and extracellular conditions is essential for cellular homeostasis and survival. The purpose of this study is to elaborate a theoretical model of cell volume homeostasis and to apply it to a simulation of human aqueous humor (AH) production. The model assumes a cell with a spherical shape and only radial deformation satisfying the property that the cell volume in rest conditions equals that of the cell couplets constituting the ciliary epithelium of the human eye. The cytoplasm is described as a homogeneous mixture containing fluid, ions, and neutral solutes whose evolution is determined by net production mechanisms occurring in the intracellular volume and by water and solute exchange across the membrane. Averaging the balance equations over the cell volume leads to a coupled system of nonlinear ordinary differential equations (ODEs) which are solved using the θ-method and the Matlab function ode15s. Simulation tests are conducted to characterize the set of parameters corresponding to baseline conditions in AH production. The model is subsequently used to investigate the relative importance of (a) impermeant charged proteins; (b) sodium–potassium (Na+/K+) pumps; (c) carbonic anhydrase (CA) in the AH production process; and (d) intraocular pressure. Results suggest that (a) and (b) play a role; (c) lacks significant weight, at least for low carbon dioxide values; and (d) plays a role for the elevated values of intraocular pressure. Model results describe a higher impact from charged proteins and Na+/K+ ATPase than CA on AH production and cellular volume. The computational virtual laboratory provides a method to further test in vivo experiments and machine learning-based data analysis toward the prevention and cure of ocular diseases such as glaucoma. 

Recent data across the globe indicates a decline in stem competency among secondary education students. Despite persistent interest in STEM fields this decrease in preparedness could yield detrimental effects for both future scientists and engineers. To address this current trend, a collaborative partnership between a university and high school commenced. The goal was to create an advanced experiential engineering course focused primarily on ophthalmology principles, research, and hands-on solutions. Twenty-one high school students (grades 9-12) enrolled in the course. Their objective was to investigate research questions involving ocular physiology. These ranged from surveying intraocular pressure measurement methods, examining the nature of vitreous humor properties, and investigating the inherent connection between blood flow and fluid dynamics. Furthermore, students engaged in hands-on experimentation that resulted in a hydraulics-based model which attempted to link the correlation between blood pressure and intraocular pressure involved in glaucoma progression. Post-course interviews revealed three major themes: i) an increased appreciation for the utility of mathematics and its real-world use; ii) the importance of the mentor-mentee relationship and professional networking; and iii) increased access to resources beyond what is traditionally found in a high school classroom. These findings suggest that incorporating research into a high school classroom can foster positive outcomes and spark students’ interest in ophthalmology research and in STEM more broadly. This course can serve as a model in future development of project-based engineering curriculum and help broaden participation in STEM.