Fructooligosaccharides act on the gut-bone axis to improve bone independent of Tregs and alter osteocytes in young adult C57BL/6 female mice

Targeting the gut-bone axis with probiotics and prebiotics is considered as a promising strategy to reduce the risk of osteoporosis. Gut-derived short chain fatty acids (SCFA) mediate the effects of probiotics on bone via Tregs, but it is not known whether prebiotics act through a similar mechanism. We investigated how 2 different prebiotics, tart cherry (TC) and fructooligosaccharide (FOS), affect bone, and whether Tregs are required for this response. Eight-wk-old C57BL/6 female mice were fed with diets supplemented with 10% w/w TC, FOS, or a control diet (Con; AIN-93M) diet, and they received an isotype control or CD25 Ab to suppress Tregs. The FOS diet increased BMC, density, and trabecular bone volume in the vertebra (~40%) and proximal tibia (~30%) compared to the TC and control diets (Con), irrespective of CD25 treatment. Both prebiotics increased (P < .01) fecal SCFAs, but the response was greater with FOS. To determine how FOS affected bone cells, we examined genes involved in osteoblast and osteoclast differentiation and activity as well as genes expressed by osteocytes. The FOS increased the expression of regulators of osteoblast differentiation (bone morphogenetic protein 2 [Bmp2], Wnt family member 10b [Wnt10b] and Osterix [Osx]) and type 1 collagen). Osteoclasts regulators were unaltered. The FOS also increased the expression of genes associated with osteocytes, including (Phex), matrix extracellular phosphoglycoprotein (Mepe), and dentin matrix acidic phosphoprotein 1 (Dmp-1). However, Sost, the gene that encodes for sclerostin was also increased by FOS as the number and density of osteocytes increased. These findings demonstrate that FOS has a greater effect on the bone mass and structure in young adult female mice than TC and that its influence on osteoblasts and osteocytes is not dependent on Tregs.

Cross-Domain Text Mining of Pathophysiological Processes Associated with Diabetic Kidney Disease

Diabetic kidney disease (DKD) is the leading cause of end-stage renal disease worldwide. This study's goal was to identify the signaling drivers and pathways that modulate glomerular endothelial dysfunction in DKD via artificial intelligence-enabled literature-based discovery. Cross-domain text mining of 33+ million PubMed articles was performed with SemNet 2.0 to identify and rank multi-scalar and multi-factorial pathophysiological concepts related to DKD. A set of identified relevant genes and proteins that regulate different pathological events associated with DKD were analyzed and ranked using normalized mean HeteSim scores. High-ranking genes and proteins intersected three domains-DKD, the immune response, and glomerular endothelial cells. The top 10% of ranked concepts were mapped to the following biological functions: angiogenesis, apoptotic processes, cell adhesion, chemotaxis, growth factor signaling, vascular permeability, the nitric oxide response, oxidative stress, the cytokine response, macrophage signaling, NFκB factor activity, the TLR pathway, glucose metabolism, the inflammatory response, the ERK/MAPK signaling response, the JAK/STAT pathway, the T-cell-mediated response, the WNT/β-catenin pathway, the renin-angiotensin system, and NADPH oxidase activity. High-ranking genes and proteins were used to generate a protein-protein interaction network. The study results prioritized interactions or molecules involved in dysregulated signaling in DKD, which can be further assessed through biochemical network models or experiments.

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.

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.

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-β). The TGF-β-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-β 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-β, 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-β, and the duration of TGF-β 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-β sources were active for longer times. We validated our model by comparing simulated dynamics for TGF-β, collagen area fraction, and macrophage cell population with independent experimental data from mouse models. Our results showed that partial removal of TGF-β sources changed the fibrotic patterns; in the presence of persistent TGF-β sources, partial removal of TGF-β 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-β sources may find applications in the current clinical trials targeting TGF-β for the resolution of lung fibrosis.

Experimental and mathematical approaches for drug delivery for the treatment of wet age-related macular degeneration

Several chronic eye diseases affect the posterior segment of the eye. Among them age-related macular degeneration can cause vision loss if left untreated and is one of the leading causes of visual impairment in the world. Most treatments are based on intravitreally injected therapeutics that inhibit the action of vascular endothelial growth factor. However, due to the need for monthly injections, this method is associated with poor patient compliance. To address this problem, numerous drug delivery systems (DDSs) have been developed. This review covers a selection of particulate systems, non-stimuli responsive hydrogels, implants, and composite systems that have been developed in the last few decades. Depending on the type of DDS, polymer material, and preparation method, different mechanical properties and drug release profiles can be achieved. Furthermore, DDS development can be optimized by implementing mathematical modeling of both drug release and pharmacokinetic aspects. Several existing mathematical models for diffusion-controlled, swelling-controlled, and erosion-controlled drug delivery from polymeric systems are summarized. Compartmental and physiologically based models for ocular drug transport and pharmacokinetics that have studied drug concentration profiles after intravitreal delivery or release from a DDS are also reviewed. The coupling of drug release models with ocular pharmacokinetic models can lead to obtaining much more efficient DDSs for the treatment of age-related macular degeneration and other diseases of the posterior segment of the eye.

Computational fluid dynamics modeling of aerosol particle transport through lung airway mucosa

Delivery of aerosols to the lung can treat various lung diseases. However, the conducting airways are coated by a protective mucus layer with complex properties that make this form of delivery difficult. Mucus is a non-Newtonian fluid and is cleared from the lungs over time by ciliated cells. Further, its gel-like structure hinders the diffusion of particles through it. Any aerosolized treatment of lung diseases must penetrate the mucosal barrier. Using computational fluid dynamics, a model of the airway mucus and periciliary layer was constructed to simulate the transport of impacted aerosol particles. The model predicts the dosage fraction of particles of a certain size that penetrate the mucus and reach the underlying tissue, as well as the distance downstream of the dosage site where tissue concentration is maximized. Reactions that may occur in the mucus are also considered, with simulated data for the interaction of a model virus and an antibody.

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-β). The TGF-β-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-β 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-β, 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-β, and the duration of TGF-β 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-β sources were active for longer times. We validated our model by comparing simulated dynamics for TGF-β, collagen area fraction, and macrophage cell population with independent experimental data from mouse models. Our results showed that partial removal of TGF-β sources changed the fibrotic patterns; in the presence of persistent TGF-β sources, partial removal of TGF-β 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-β sources may find applications in the current clinical trials targeting TGF-β 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-β as a critical regulator of fibrosis. We built a model to investigate the mechanisms of TGF-β sources in the process of fibrosis. Our results showed spatial behavior of sources (stationary or mobile) and their activity (activation rate of TGF-β, longer activation of sources) could lead to lung fibrosis. Current clinical trials for fibrosis that target TGF-β need to consider TGF-β sources' spatial properties and activity to develop better treatment strategies.

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.

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 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.

Pathophysiology of mesangial expansion in diabetic nephropathy: mesangial structure, glomerular biomechanics, and biochemical signaling and regulation

Diabetic nephropathy, a kidney complication arising from diabetes, is the leading cause of death in diabetic patients. Unabated, the growing epidemic of diabetes is increasing instances of diabetic nephropathy. Although the main causes of diabetic nephropathy have been determined, the mechanisms of their combined effects on cellular and tissue function are not fully established. One of many damages of diabetic nephropathy is the development of fibrosis within the kidneys, termed mesangial expansion. Mesangial expansion is an important structural lesion that is characterized by the aberrant proliferation of mesangial cells and excess production of matrix proteins. Mesangial expansion is involved in the progression of kidney failure in diabetic nephropathy, yet its causes and mechanism of impact on kidney function are not well defined. Here, we review the literature on the causes of mesangial expansion and its impacts on cell and tissue function. We highlight the gaps that still remain and the potential areas where bioengineering studies can bring insight to mesangial expansion in diabetic nephropathy.

Mathematical Modeling of the Gut-Bone Axis and Implications of Butyrate Treatment on Osteoimmunology

Butyrate, a short-chain fatty acid produced by the gut microbiota, has pivotal roles in the regulation of the immune system. Recent studies have revealed that butyrate increases the differentiation of peripheral regulatory T cells in the gut-bone axis and promotes osteoblasts' bone forming activity. However, the mechanism of the therapeutic benefit of butyrate in bone remodeling remains incompletely understood. Here, we develop a multicompartment mathematical model to quantitatively predict the contribution of butyrate on the expansion of regulatory T cells in the gut, blood, and bone compartments. We investigate the interplay between regulatory T cell-derived TGF-β and CD8+ T cell-derived Wnt-10b with changes in gut butyrate concentration. In addition, we connect our model to a detailed model of bone metabolism to study the impacts of butyrate and Wnt-10b on trabecular bone volume. Our results indicate both direct and indirect immune-mediated impacts of butyrate on bone metabolism.

Iterative community-driven development of a SARS-CoV-2 tissue simulator

The 2019 novel coronavirus, SARS-CoV-2, is a pathogen of critical significance to international public health. Knowledge of the interplay between molecular-scale virus-receptor interactions, single-cell viral replication, intracellular-scale viral transport, and emergent tissue-scale viral propagation is limited. Moreover, little is known about immune system-virus-tissue interactions and how these can result in low-level (asymptomatic) infections in some cases and acute respiratory distress syndrome (ARDS) in others, particularly with respect to presentation in different age groups or pre-existing inflammatory risk factors. Given the nonlinear interactions within and among each of these processes, multiscale simulation models can shed light on the emergent dynamics that lead to divergent outcomes, identify actionable "choke points" for pharmacologic interventions, screen potential therapies, and identify potential biomarkers that differentiate patient outcomes. Given the complexity of the problem and the acute need for an actionable model to guide therapy discovery and optimization, we introduce and iteratively refine a prototype of a multiscale model of SARS-CoV-2 dynamics in lung tissue. The first prototype model was built and shared internationally as open source code and an online interactive model in under 12 hours, and community domain expertise is driving regular refinements. In a sustained community effort, this consortium is integrating data and expertise across virology, immunology, mathematical biology, quantitative systems physiology, cloud and high performance computing, and other domains to accelerate our response to this critical threat to international health. More broadly, this effort is creating a reusable, modular framework for studying viral replication and immune response in tissues, which can also potentially be adapted to related problems in immunology and immunotherapy.

Multiscale modeling in disease

Multiscale computational modeling aims to connect the complex networks of effects at different length and/or time scales. For example, these networks often include intracellular molecular signaling, crosstalk, and other interactions between neighboring cell populations, and higher levels of emergent phenomena across different regions of tissues and among collections of tissues or organs interacting with each other in the whole body. Recent applications of multiscale modeling across intracellular, cellular, and/or tissue levels are highlighted here. These models incorporated the roles of biochemical and biomechanical modulation in processes that are implicated in the mechanisms of several diseases including fibrosis, joint and bone diseases, respiratory infectious diseases, and cancers.

Neonicotinoid exposure disrupts bumblebee nest behavior, social networks, and thermoregulation

Neonicotinoid pesticides can negatively affect bee colonies, but the behavioral mechanisms by which these compounds impair colony growth remain unclear. Here, we investigate imidacloprid's effects on bumblebee worker behavior within the nest, using an automated, robotic platform for continuous, multicolony monitoring of uniquely identified workers. We find that exposure to field-realistic levels of imidacloprid impairs nursing and alters social and spatial dynamics within nests, but that these effects vary substantially with time of day. In the field, imidacloprid impairs colony thermoregulation, including the construction of an insulating wax canopy. Our results show that neonicotinoids induce widespread disruption of within-nest worker behavior that may contribute to impaired growth, highlighting the potential of automated techniques for characterizing the multifaceted, dynamic impacts of stressors on behavior in bee colonies.

Neonicotinoid exposure disrupts bumblebee nest behavior, social networks, and thermoregulation

Neonicotinoid pesticides can negatively affect bee colonies, but the behavioral mechanisms by which these compounds impair colony growth remain unclear. Here, we investigate imidacloprid’s effects on bumblebee worker behavior within the nest, using an automated, robotic platform for continuous, multicolony monitoring of uniquely identified workers. We find that exposure to field-realistic levels of imidacloprid impairs nursing and alters social and spatial dynamics within nests, but that these effects vary substantially with time of day. In the field, imidacloprid impairs colony thermoregulation, including the construction of an insulating wax canopy. Our results show that neonicotinoids induce widespread disruption of within-nest worker behavior that may contribute to impaired growth, highlighting the potential of automated techniques for characterizing the multifaceted, dynamic impacts of stressors on behavior in bee colonies.

Mathematical Modeling of Tuberculosis Granuloma Activation

Tuberculosis (TB) is one of the most common infectious diseases worldwide. It is estimated that one-third of the world’s population is infected with TB. Most have the latent stage of the disease that can later transition to active TB disease. TB is spread by aerosol droplets containing Mycobacterium tuberculosis (Mtb). Mtb bacteria enter through the respiratory system and are attacked by the immune system in the lungs. The bacteria are clustered and contained by macrophages into cellular aggregates called granulomas. These granulomas can hold the bacteria dormant for long periods of time in latent TB. The bacteria can be perturbed from latency to active TB disease in a process called granuloma activation when the granulomas are compromised by other immune response events in a host, such as HIV, cancer, or aging. Dysregulation of matrix metalloproteinase 1 (MMP-1) has been recently implicated in granuloma activation through experimental studies, but the mechanism is not well understood. Animal and human studies currently cannot probe the dynamics of activation, so a computational model is developed to fill this gap. This dynamic mathematical model focuses specifically on the latent to active transition after the initial immune response has successfully formed a granuloma. Bacterial leakage from latent granulomas is successfully simulated in response to the MMP-1 dynamics under several scenarios for granuloma activation.

Derivation of an Analytical Solution to a Reaction-Diffusion Model for Autocatalytic Degradation and Erosion in Polymer Microspheres

A mathematical reaction-diffusion model is defined to describe the gradual decomposition of polymer microspheres composed of poly(D,L-lactic-co-glycolic acid) (PLGA) that are used for pharmaceutical drug delivery over extended periods of time. The partial differential equation (PDE) model treats simultaneous first-order generation due to chemical reaction and diffusion of reaction products in spherical geometry to capture the microsphere-size-dependent effects of autocatalysis on PLGA erosion that occurs when the microspheres are exposed to aqueous media such as biological fluids. The model is solved analytically for the concentration of the autocatalytic carboxylic acid end groups of the polymer chains that comprise the microspheres as a function of radial position and time. The analytical solution for the reaction and transport of the autocatalytic chemical species is useful for predicting the conditions under which drug release from PLGA microspheres transitions from diffusion-controlled to erosion-controlled release, for understanding the dynamic coupling between the PLGA degradation and erosion mechanisms, and for designing drug release particles. The model is the first to provide an analytical prediction for the dynamics and spatial heterogeneities of PLGA degradation and erosion within a spherical particle. The analytical solution is applicable to other spherical systems with simultaneous diffusive transport and first-order generation by reaction.

Bifurcation study of blood flow control in the kidney

Renal blood flow is maintained within a narrow window by a set of intrinsic autoregulatory mechanisms. Here, a mathematical model of renal hemodynamics control in the rat kidney is used to understand the interactions between two major renal autoregulatory mechanisms: the myogenic response and tubuloglomerular feedback. A bifurcation analysis of the model equations is performed to assess the effects of the delay and sensitivity of the feedback system and the time constants governing the response of vessel diameter and smooth muscle tone. The results of the bifurcation analysis are verified using numerical simulations of the full nonlinear model. Both the analytical and numerical results predict the generation of limit cycle oscillations under certain physiologically relevant conditions, as observed in vivo.

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.

Optimal Control of One-dimensional Cellular Uptake in Tissue Engineering

A control problem motivated by tissue engineering is formulated and solved in which control of the uptake of growth factors (signaling molecules) is necessary to spatially and temporally regulate cellular processes for the desired growth or regeneration of a tissue. Four approaches are compared for determining 1D optimal boundary control trajectories for a distributed parameter model with reaction, diffusion, and convection: (i) basis function expansion, (ii) method of moments, (iii) internal model control (IMC), and (iv) model predictive control (MPC). The proposed method-of-moments approach is computationally efficient while enforcing a non-negativity constraint on the control input. While more computationally expensive than methods (i)-(iii), the MPC formulation significantly reduced the computational cost compared to simultaneous optimization of the entire control trajectory. A comparison of the pros and cons of each of the four approaches suggests that an algorithm that combines multiple approaches is most promising for solving the optimal control problem for multiple spatial dimensions.

Mathematical modeling of drug delivery from autocatalytically degradable PLGA microspheres--a review

PLGA microspheres are widely studied for controlled release drug delivery applications, and many models have been proposed to describe PLGA degradation and erosion and drug release from the bulk polymer. Autocatalysis is known to have a complex role in the dynamics of PLGA erosion and drug transport and can lead to size-dependent heterogeneities in otherwise uniformly bulk-eroding polymer microspheres. The aim of this review is to highlight mechanistic, mathematical models for drug release from PLGA microspheres that specifically address interactions between phenomena generally attributed to autocatalytic hydrolysis and mass transfer limitation effects. Predictions of drug release profiles by mechanistic models are useful for understanding mechanisms and designing drug release particles.