Energy-based operator splitting approach for the time discretization of coupled systems of partial and ordinary differential equations for fluid flows: The Stokes case

The goal of this work is to develop a novel splitting approach for the numerical solution of multiscale problems involving the coupling between Stokes equations and ODE systems, as often encountered in blood flow modeling applications. The proposed algorithm is based on a semi-discretization in time based on operator splitting, whose design is guided by the rationale of ensuring that the physical energy balance is maintained at the discrete level. As a result, unconditional stability with respect to the time step choice is ensured by the implicit treatment of interface conditions within the Stokes substeps, whereas the coupling between Stokes and ODE substeps is enforced via appropriate initial conditions for each substep. Notably, unconditional stability is attained without the need of subiterating between Stokes and ODE substeps. Stability and convergence properties of the proposed algorithm are tested on three specific examples for which analytical solutions are derived

A Multiscale Thermo-Fluid Computational Model for a Two-Phase Cooling System

In this paper, we describe a mathematical model and a numerical simulation method for the condenser component of a novel two-phase thermosyphon cooling system for power electronics applications. The condenser consists of a set of roll-bonded vertically mounted fins among which air flows by either natural or forced convection. In order to deepen the understanding of the mechanisms that determine the performance of the condenser and to facilitate the further optimization of its industrial design, a multiscale approach is developed to reduce as much as possible the complexity of the simulation code while maintaining reasonable predictive accuracy. To this end, heat diffusion in the fins and its convective transport in air are modeled as 2D processes while the flow of the two-phase coolant within the fins is modeled as a 1D network of pipes. For the numerical solution of the resulting equations, a Dual Mixed-Finite Volume scheme with Exponential Fitting stabilization is used for 2D heat diffusion and convection while a Primal Mixed Finite Element discretization method with upwind stabilization is used for the 1D coolant flow. The mathematical model and the numerical method are validated through extensive simulations of realistic device structures which prove to be in excellent agreement with available experimental data

Multiscale mathematical modeling of ocular blood flow and oxygenation and their relevance to glaucoma

Glaucoma is a multifactorial ocular disease progressively leading to irreversible blindness. There is clear evidence of correlations between alterations in ocular hemodynamics and glaucoma; however, the mechanisms giving rise to these correlations are still elusive. The objective of this thesis is to develop mathematical models and methods to help elucidate these mechanisms. First, we develop a mathematical model that describes the deformation of ocular structures and ocular blood flow using a reduced-order fluid-structure interaction model. This model is used to investigate the relevance of mechanical and vascular factors in glaucoma. As a first step in expanding this model to higher dimensions, we propose a novel energy-based technique for coupling partial and ordinary differential equations in blood flow, using operator splitting. Next, we combine clinical data and model predictions to propose possible explanations for the increase in venous oxygen saturation in advanced glaucoma patients. We develop a computer-aided manipulation process of color Doppler images to extract novel waveform parameters to distinguish between healthy and glaucomatous individuals. The results obtained in this work suggest that: 1) the increase in resistance of the retinal microcirculation contributes to the influence of intraocular pressure on retinal hemodynamics; 2) the influence of cerebrospinal fluid pressure on retinal hemodynamics is mediated by associated changes in blood pressure; 3) the increase in venous oxygen saturation levels observed among advanced glaucoma patients depends on the value of the patients’ intraocular pressure; 4) the normalized distance between the ascending and descending limb of the ophthalmic artery velocity profile is significantly higher in glaucoma patients than in healthy individuals

Physiology-Enhanced Data Analytics to Evaluate the Effect of Altitude on Intraocular Pressure and Ocular Hemodynamics

Altitude affects intraocular pressure (IOP); however, the underlying mechanisms involved and its relationship with ocular hemodynamics remain unknown. Herein, a validated mathematical modeling approach was used for a physiology-enhanced (pe-) analysis of the Mont Blanc study (MBS), estimating the effects of altitude on IOP, blood pressure (BP), and retinal hemodynamics. In the MBS, IOP and BP were measured in 33 healthy volunteers at 77 and 3466 m above sea level. Pe-retinal hemodynamics analysis predicted a statistically significant increase (p < 0.001) in the model predicted blood flow and pressure within the retinal vasculature following increases in systemic BP with altitude measured in the MBS. Decreased IOP with altitude led to a non-monotonic behavior of the model predicted retinal vascular resistances, with significant decreases in the resistance of the central retinal artery (p < 0.001) and retinal venules (p = 0.003) and a non-significant increase in the resistance in the central retinal vein (p = 0.253). Pe-aqueous humor analysis showed that a decrease in osmotic pressure difference (OPD) may underlie the difference in IOP measured at different altitudes in the MBS. Our analysis suggests that venules bear the significant portion of the IOP pressure load within the ocular vasculature, and that OPD plays an important role in regulating IOP with changes in altitude

Computer-aided ophthalmic artery waveform analysis in healthy individuals and glaucoma patients

Purpose: Arterial waveform parameters (WPs) are commonly used to monitor and diagnose systemic diseases. Color Doppler Imaging (CDI) is a consolidated technique to measure blood velocity profile in some of the major ocular vessels. This study proposes a computer-aided manipulation process of ophthalmic artery (OA) CDI images to classify and quantify WPs that might be significant in the assessment of glaucoma. Methods: Fifty CDI images acquired by four different operators on nine healthy individuals and 38 CDI images of 38 open-angle glaucoma (OAG) patients were considered. An ad-hoc semi-automated image processing code was implemented to detect the digitalized OA velocity waveform and to extract the WPs. Concordance correlation coefficient (CCC), two-sample t-test and Pearson\u2019s correlation coefficient were used to test for similarities, differences and associations among variables

Computer-aided identification of novel ophthalmic artery waveform parameters in healthy subjects and glaucoma patients

Arterial waveform parameters (WPs) are commonly used to monitor and diagnose systemic diseases. Color Doppler Imaging (CDI) is a consolidated technique to measure blood velocity profile in some of the major ocular vessels. This study proposes a computer-aided manipulation process of ophthalmic artery (OA) CDI images to classify and quantify WPs that might be significant in the assessment of glaucom