Analysis of Finite Difference Discretization Schemes for Diffusion in Spheres with Variable Diffusivity.

Two finite difference discretization schemes for approximating the spatial derivatives in the diffusion equation in spherical coordinates with variable diffusivity are presented and analyzed. The numerical solutions obtained by the discretization schemes are compared for five cases of the functional form for the variable diffusivity: (I) constant diffusivity, (II) temporally-dependent diffusivity, (III) spatially-dependent diffusivity, (IV) concentration-dependent diffusivity, and (V) implicitly-defined, temporally- and spatially-dependent diffusivity. Although the schemes have similar agreement to known analytical or semi-analytical solutions in the first four cases, in the fifth case for the variable diffusivity, one scheme produces a stable, physically reasonable solution, while the other diverges. We recommend the adoption of the more accurate and stable of these finite difference discretization schemes to numerically approximate the spatial derivatives of the diffusion equation in spherical coordinates for any functional form of variable diffusivity, especially cases where the diffusivity is a function of position.

Computer Modeling of Bevacizumab Drug Distribution After Intravitreal Injection in Rabbit and Human Eyes.

Age-related macular degeneration (AMD) is a condition brought on by macular deterioration caused primarily by inflammation and cell death in the retina. There is no cure for the disease and current treatments for advanced (wet) AMD rely on intravitreal injections of anti-vascular endothelial growth factor (anti-VEGF) therapeutics. One common off-label anti-VEGF drug used in AMD treatment is bevacizumab. There have been experimental efforts to investigate the pharmacokinetic (PK) behavior of bevacizumab in the vitreous and aqueous humor. Still the quantitative effect of elimination routes and drug concentration in the macula are not well understood. In our study, we developed two spatial models representing rabbit and human vitreous humor to better understand the PK behavior of bevacizumab. We explored convective effects on the vitreous while considering the anterior elimination alone or coupled with posterior elimination. We compared our models with available experimental data and calculated an approximate macula concentration. Our results show that both anterior and posterior elimination play a role in bevacizumab clearance from the eye. Furthermore, an effective bevacizumab concentration close to the macula region is maintained for shorter time periods when compared to the whole vitreous region. This model can improve knowledge and understanding of AMD treatment.

Mathematical Modeling of Drug Delivery from Bi-Layered Core-Shell Polymeric Microspheres.

Chronic diseases usually require repetitive dosing. Depending on factors such as dosing frequency, mode of administration, and associated costs this can result in poor patient compliance. A better alternative involves using drug delivery systems to reduce the frequency of dosing and extend drug release. However, reaching the market stage is a time-consuming process. In this study, we used two numerical approaches for estimating the values of the critical parameters that govern the diffusion-controlled drug release within bilayered core-shell microspheres. Specifically, the estimated parameters include burst release, drug diffusion coefficient in two polymers, and the drug partition coefficient. Estimating these parameters provides insight for optimizing device design, guiding experimental efforts, and improving the device's effectiveness. We obtained good agreement between the models and the experimental data. The methods explored in this work apply not only to bi-layered spherical systems but can also be extended to multi-layered spherical systems.

Logic-Based Modeling of Inflammatory Macrophage Crosstalk with Glomerular Endothelial Cells in Diabetic Kidney Disease.

Diabetic kidney disease is a complication in 1 out of 3 patients with diabetes. Aberrant glucose metabolism in diabetes leads to an immune response causing inflammation and to structural and functional damage in the glomerular cells of the kidney. Complex cellular signaling lies at the core of metabolic and functional derangement. Unfortunately, the mechanism underlying the role of inflammation in glomerular endothelial cell dysfunction during diabetic kidney disease is not fully understood. Computational models in systems biology allow the integration of experimental evidence and cellular signaling networks to understand mechanisms involved in disease progression. We built a logic-based ordinary differential equations model to study macrophage-dependent inflammation in glomerular endothelial cells during diabetic kidney disease progression. We studied the crosstalk between macrophages and glomerular endothelial cells in the kidney using a protein signaling network stimulated with glucose and lipopolysaccharide. The network and model were built using the open-source software package Netflux. This modeling approach overcomes the complexity of studying network models and the need for extensive mechanistic details. The model simulations were fitted and validated against available biochemical data from in vitro experiments. The model identified mechanisms responsible for dysregulated signaling in macrophages and glomerular endothelial cells during diabetic kidney disease. In addition, we investigated the influence of signaling interactions and species that on glomerular endothelial cell morphology through selective knockdown and downregulation. We found that partial knockdown of VEGF receptor 1, PLC-γ, adherens junction proteins, and calcium partially recovered the endothelial cell fenestration size. Our model findings contribute to understanding signaling and molecular perturbations that affect the glomerular endothelial cells in the early stage of diabetic kidney disease.

An agent-based modeling approach for lung fibrosis in response to COVID-19.

The severity of the COVID-19 pandemic has created an emerging need to investigate the long-term effects of infection on patients. Many individuals are at risk of suffering pulmonary fibrosis due to the pathogenesis of lung injury and impairment in the healing mechanism. Fibroblasts are the central mediators of extracellular matrix (ECM) deposition during tissue regeneration, regulated by anti-inflammatory cytokines including transforming growth factor beta (TGF-ß). The TGF-ß-dependent accumulation of fibroblasts at the damaged site and excess fibrillar collagen deposition lead to fibrosis. We developed an open-source, multiscale tissue simulator to investigate the role of TGF-ß sources in the progression of lung fibrosis after SARS-CoV-2 exposure, intracellular viral replication, infection of epithelial cells, and host immune response. Using the model, we predicted the dynamics of fibroblasts, TGF-ß, and collagen deposition for 15 days post-infection in virtual lung tissue. Our results showed variation in collagen area fractions between 2% and 40% depending on the spatial behavior of the sources (stationary or mobile), the rate of activation of TGF-ß, and the duration of TGF-ß sources. We identified M2 macrophages as primary contributors to higher collagen area fraction. Our simulation results also predicted fibrotic outcomes even with lower collagen area fraction when spatially-localized latent TGF-ß sources were active for longer times. We validated our model by comparing simulated dynamics for TGF-ß, collagen area fraction, and macrophage cell population with independent experimental data from mouse models. Our results showed that partial removal of TGF-ß sources changed the fibrotic patterns; in the presence of persistent TGF-ß sources, partial removal of TGF-ß from the ECM significantly increased collagen area fraction due to maintenance of chemotactic gradients driving fibroblast movement. The computational findings are consistent with independent experimental and clinical observations of collagen area fractions and cell population dynamics not used in developing the model. These critical insights into the activity of TGF-ß sources may find applications in the current clinical trials targeting TGF-ß for the resolution of lung fibrosis. Author summary: COVID-19 survivors are at risk of lung fibrosis as a long-term effect. Lung fibrosis is the excess deposition of tissue materials in the lung that hinder gas exchange and can collapse the whole organ. We identified TGF-ß as a critical regulator of fibrosis. We built a model to investigate the mechanisms of TGF-ß sources in the process of fibrosis. Our results showed spatial behavior of sources (stationary or mobile) and their activity (activation rate of TGF-ß, longer activation of sources) could lead to lung fibrosis. Current clinical trials for fibrosis that target TGF-ß need to consider TGF-ß sources' spatial properties and activity to develop better treatment strategies.

Mathematical Modeling of Impacts of Patient Differences on COVID-19 Lung Fibrosis Outcomes.

Patient-specific premorbidity, age, and sex are significant heterogeneous factors that influence the severe manifestation of lung diseases, including COVID-19 fibrosis. The renin-angiotensin system (RAS) plays a prominent role in regulating effects of these factors. Recent evidence suggests that patient-specific alteration of RAS homeostasis with premorbidity and the expression level of angiotensin converting enzyme 2 (ACE2), depending on age and sex, is correlated with lung fibrosis. However, conflicting evidence suggests decreases, increases, or no changes in RAS after SARS-CoV-2 infection. In addition, detailed mechanisms connecting the patient-specific conditions before infection to infection-induced fibrosis are still unknown. Here, a mathematical model is developed to quantify the systemic contribution of heterogeneous factors of RAS in the progression of lung fibrosis. Three submodels are connected-a RAS model, an agent-based COVID-19 in-host immune response model, and a fibrosis model-to investigate the effects of patient-group-specific factors in the systemic alteration of RAS and collagen deposition in the lung. The model results indicate cell death due to inflammatory response as a major contributor to the reduction of ACE and ACE2, whereas there are no significant changes in ACE2 dynamics due to viral-bound internalization of ACE2. Reduction of ACE reduces the homeostasis of RAS including angiotensin II (ANGII), while the decrease in ACE2 increases ANGII and results in severe lung injury and fibrosis. The model explains possible mechanisms for conflicting evidence of RAS alterations in previously published studies. Also, the results show that ACE2 variations with age and sex significantly alter RAS peptides and lead to fibrosis with around 20% additional collagen deposition from systemic RAS with slight variations depending on age and sex. This model may find further applications in patient-specific calibrations of tissue models for acute and chronic lung diseases to develop personalized treatments.

Mathematical modeling of the effects of Wnt-10b on bone metabolism.

Bone health is determined by factors including bone metabolism or remodeling. Wnt-10b alters osteoblastogenesis through pre-osteoblast proliferation and differentiation and osteoblast apoptosis rate, which collectively lead to the increase of bone density. To model this, we adapted a previously published model of bone remodeling. The resulting model for the bone compartment includes differential equations for active osteoclasts, pre-osteoblasts, osteoblasts, osteocytes, and the amount of bone present at the remodeling site. Our alterations to the original model consist of extending it past a single remodeling cycle and implementing a direct relationship to Wnt-10b. Four new parameters were estimated and validated using normalized data from mice. The model connects Wnt-10b to bone metabolism and predicts the change in trabecular bone volume caused by a change in Wnt-10b input. We find that this model predicts the expected increase in pre-osteoblasts and osteoblasts while also pointing to a decrease in osteoclasts when Wnt-10b is increased.