Many mathematical models in biology and physiology are represented by systems of nonlinear differential equations. In recent years these models have become increasingly large-scale and multiphysics, as increasing amounts of data are available on the properties and behaviour of biological systems. Often an observed behaviour of interest in a model may be written as a linear functional. A key question therefore is to determine which terms in the model have the greatest effect on functionals of interest. An approach for answering this question has recently been developed, called model reduction using a posteriori analysis. The method was initially developed for systems of nonlinear initial value ordinary differential equations, and automatically identifies regions of the computational domain and components of the model solution where an accurate mathematical representation of the model is required to accurately calculate a linear functional of interest. Initial-value ordinary differential equations can be written as a first-order derivative term plus an algebraic 'reaction' term. In previous work on model reduction using a posteriori analysis the algebraic 'reaction' term is removed from the equations in the reduced model. The first contribution of this thesis is to extend the method so that the first-order derivative term is removed from the differential equation instead of the algebraic 'reaction' term, resulting in a quasi-steady state approximation in automatically identified regions of the domain and components of the solution. The second contribution of this thesis is to extend the method to boundary value problems and partial differential equations. We consider differential equations with algebraic terms, first order terms and second order terms, any combination of which may be nonlinear. The method is used to automatically simplify several differential equation models including models of chemotaxis and tissue-level cardiac electrophysiology.

Abstract Background – The heart cell is a prime example of a system, in which numerous interconnected regulatory mechanisms affect the dynamic balance of cellular function. The function of the system emerges from the interactions of its components rather than from their individual properties. Thus, it is a challenging task to understand the causal relations within such a system, based on the analysis of experimental results. Facing this complexity, the systems biological approach has gained interest during recent years, since with using it we can make an effort to observe, quantitatively and simultaneously, multiple components and their interdependencies in biological networks. Methods and aims – One of the most important tools in systems biology is mathematical modelling. In this thesis, novel model components have been developed and existing components integrated to describe mathematically the calcium dynamics in cardiac myocytes with improved physiological accuracy. Special attention was paid to both the activity-dependent and automatic regulation of the dynamics. This enabled the quantitative analysis of the regulation’s role in both physiological and pathophysiological conditions. Results – Validation of the novel model components that describe the calcium transport mechanisms indicates that the developed schemes are accurate and applicable also beyond the normal physiological state of the cardiac myocyte. Results also highlight the importance of autoregulation of calcium dynamics in the excitation-contraction coupling. Furthermore, the analysis indicates that the CaMK-dependent regulation of the calcium uptake to and release from the sarcoplasmic reticulum calcium stores could have substantial roles as downstream effectors in beta-adrenergic stimulation. Conclusions – Results emphasize mathematical modelling as a valuable complement to experiments in understanding causal relations within complex biological systems such as the cardiac myocytes. That is, rigorous data integration with mathematical models can provide significant insight to the quantitative role of both the individual model components and the interconnected regulatory loops. This is especially true for the analysis of genetically engineered animal models, in which the intended modification is always accompanied by compensatory changes that can mask to a varying degree the actual phenomenon of interest.

<p dir="ltr">The vaginal microbiome and female sex hormones are both involved in the development and progression of gynecological pathologies. The individual mechanisms by which the vaginal microbiome leads to disease progression and how female sex hormones are known. However, the mechanisms by which the vaginal microbiome regulates female sex hormones, such as progesterone, are not well understood. This study seeks to understand how the vaginal microbiome regulates progesterone receptor (PGR) expression via secondary analysis of host and vaginal microbiome multi-omics data from the Partners PrEP cohort. This dataset consists of cervicovaginal samples of women enrolled in the Partners PrEP study. Partial Least Squares Regression (PLSR) models were created for each biological data type (microbial composition, metabolomics, metaproteomics) to assess how these factors regulate PGR expression. Significant factors were identified through variable importance of projection (VIP) and correlation analysis. Partial correlation analysis and follow-up PLSR models incorporating clinical and demographic variables were performed to assess the robustness of the vaginal microbiome-PGR associations. The PLSR models indicated lower PGR expression was associated with <i>G. vaginalis,</i> and higher PGR expression was associated with <i>Lactobacillus </i>species. Cytosine, guanine, and tyrosine were among metabolites significantly associated with higher PGR expression and experimentally determined to be produced by <i>Lactobacillus</i> species. Conversely, citrulline and succinate were associated with lower PGR expression and experimentally determined to be produced by <i>G. vaginalis</i>. The models indicated that bacterial metabolic pathways involved in glucose metabolism, such as glucagon signaling and starch and sugar metabolism, may regulate PGR expression. Demographic phenotypes were also considered from the dataset and did not significantly alter the association between the biological explanatory variables and PGR expression. The results indicate that guanine, cytosine, succinate, starch and sucrose metabolism, and glycolysis gluconeogenesis may be regulators of PGR abundance and function. The models suggest vaginal microbiome factors could play a role in gynecological conditions where progesterone signaling is suppressed. Future experimental work is needed to validate the results of these models and support their use as predictive tools to understand the role of the vaginal microbiome.</p>

<p dir="ltr">The focus is on the complex nature of primary blast injury (PBI) and employs advanced simulation techniques to model the physiological impacts using a TBI-on-a-chip system. This study involves a two-way Fluid-Structure Interaction (FSI) model in ANSYS, coupling Transient Structural and Fluent modules to simulate the effects of a blast wave on brain tissue. The research explores the creation and validation of boundary conditions, such as fixed support and varying strain rates, to ensure the reliability of the experimental setup. Key findings include the non-uniform distribution of strain, which has significant implications for understanding injury mechanisms and inflammatory marker analysis. The project also provides a detailed workflow for FSI simulations, highlighting the advantages of uniform mechanical loading and its impact on experimental accuracy.</p>

<p>  </p> <p>Traumatic Brain Injuries (TBI) are a major cause of disability and death in the United States. One of the greatest consequences of the disease is the resulting long-term damage, especially in milder injury cases where the damage is initially subclinical and thus lacking acutely observable manifestations that over time can compound significantly. Among these chronic issues, Alzheimer’s Disease (AD) is one of the most serious. While multiple studies demonstrate an increased likelihood of developing neurodegenerative diseases in response to TBI, the underlying mechanisms remain undefined and no current treatment options are available. Multiple hypotheses have been postulated based on various animal and clinical models, which have contributed a great deal to our current knowledge base and implicated several targets of interest in this pathway (i.g. oxidative stress, inflammation, disruptions in proteostasis). While extremely valuable, these <em>in vivo</em> procedures and analyses are physiologically and ethically complex: there is currently no model capable of separating and visualizing TBI-induced sub-cellular damage in the moments (seconds) immediately following injury, and the subsequent associated long-term changes (AD). In addition, no mechanistic study has been performed to link mechanical-trauma independently with neurodegeneration initiation via protein aggregation. It is clear that additional investigative tools are needed to rectify these intricate issues, and while <em>in vitro </em>methodologies generally offer the type of resolution required, no such model replicates these phenomena. Therefore, we introduce the “TBI-on-a-chip” <em>in vitro </em>concussive model, with a series of concomitant targeted-experiments to address this urgent, currently unmet need. This dissertation work describes the development of our cellular trauma model, featuring a multi-disciplinary approach that provides investigatory opportunities into cellular mechanics, molecular biology, functional alterations (electrophysiology), and morphology, in both primary and secondary injury. Utilizing this model, we directly observe evidence of impact-induced electrical/functional and biochemical consequences, in addition to isolating oxidative stress as a key, contributing component. Taken together, these collective efforts suggest that oxidative stress may be a viable target for both acute and chronic potential therapeutic interventions.</p>

<p dir="ltr">During many biological processes, cells move through their surrounding environment by exerting mechanical forces. The mechanical forces are mainly generated by molecular interactions between actin filaments (F-actins) and myosin motors within the actin cytoskeleton. Forces are transmitted to the surrounding extracellular matrix via adhesions. In this thesis, we employed agent-based computational models to study the interactions between F-actins and myosin in the motility assay and the cell migration process. In the first project, the myosin motility assay was employed to study the collective behaviors of F-actins. Unlike most of the previous computational models, we explicitly represent myosin motors. By running simulations under various conditions, we showed how the length, bending stiffness, and concentration affect the collective behavior of F-actins. We found that four distinct structures formed: homogeneous networks, flocks, bands, and rings. In addition, we showed that mobile motors lead to the formation of distinct F-actin clusters that were not observed with immobile motors. In the second project, we developed a 3D migration model to define how cells mechanically interact with their 3D environment during migration. Unlike cells migrating on a surface, cells within 3D extracellular matrix (ECM) must remodel the ECM and/or squeeze their body through the ECM, which causes 3D cell migration to be significantly more challenging than 2D migration. Our model describes realistic structural and rheological properties of ECM, cell protrusion, and focal adhesions between cells and the ECM.</p>

This thesis investigates the high-order hierarchical finite element method, also known as the finite element p-version, as a computationally-efficient technique for generating numerical solutions to the cardiac monodomain equation. We first present it as a uniform-order method, and through an a priori error bound we explain why the associated cardiac cell model must be thought of as a PDE and approximated to high-order in order to obtain the accuracy that the p-version is capable of. We perform simulations demonstrating that the achieved error agrees very well with the a priori error bound. Further, in terms of solution accuracy for time taken to solve the linear system that arises in the finite element discretisation, it is more efficient that the state-of-the-art piecewise linear finite element method. We show that piecewise linear FEM actually introduces quite significant amounts of error into the numerical approximations, particularly in the direction perpendicular to the cardiac fibres with physiological conductivity values, and that without resorting to extremely fine meshes with elements considerably smaller than 70 micrometres, we can not use it to obtain high-accuracy solutions. In contrast, the p-version can produce extremely high accuracy solutions on meshes with elements around 300 micrometres in diameter with these conductivities. Noting that most of the numerical error is due to under-resolving the wave-front in the transmembrane potential, we also construct an adaptive high-order scheme which controls the error locally in each element by adjusting the finite element polynomial basis degree using an analytically-derived a posteriori error estimation procedure. This naturally tracks the location of the wave-front, concentrating computational effort where it is needed most and increasing computational efficiency. The scheme can be controlled by a user-defined error tolerance parameter, which sets the target error within each element as a proportion of the local magnitude of the solution as measured in the H^1 norm. This numerical scheme is tested on a variety of problems in one, two and three dimensions, and is shown to provide excellent error control properties and to be likely capable of boosting efficiency in cardiac simulation by an order of magnitude. The thesis amounts to a proof-of-concept of the increased efficiency in solving the linear system using adaptive high-order finite elements when performing single-thread cardiac simulation, and indicates that the performance of the method should be investigated in parallel, where it can also be expected to provide considerable improvement. In general, the selection of a suitable preconditioner is key to ensuring efficiency; we make use of a variety of different possibilities, including one which can be expected to scale very well in parallel, meaning that this is an excellent candidate method for increasing the efficiency of cardiac simulation using high-performance computing facilities.

<p>The fluid dynamics of blood in the systemic circulation modulates production of nitric oxide (NO), a potent vasodilator. Non-invasive techniques such as the flow-mediated dilation (FMD) test and physiologic phenomena associated with autonomic stress induce hyperemia and subsequently higher levels of wall shear stress (WSS), stimulating endothelial nitric oxide synthase (eNOS) expression. In the current clinical practice, WSS–a key regulator of endothelial function–is commonly estimated assuming a parabolic velocity distribution, despite the evidence that the temporal changes of pulsatile blood flow over the cardiac cycle modulate vasodilation in mammals. This work investigates the effect of cardiovascular stress on local WSS distributions and the potential for near-wall accumulation of nitrite, the vasoactive storage form of NO in the bloodstream. The specific aims of the project are therefore as follows: 1) develop a reduced-order model of the major systemic vasculature at rest, during a flight-or-flight response, and under moderate levels of aerobic exercise; 2) derive a velocity-driven Womersley solution for pulsatile flow to support accurate estimation of pulsatile WSS in the clinical setting; and 3) quantify cumulative transport of nitrite in a multiscale model of bifurcating vasculature utilizing computational fluid dynamics (CFD). Development of these open-source, translatable methods enable accurate quantification of hemodynamics and species transport during cardiovascular stress. Results detailed herein extend our knowledge about regulation of regional blood flow during autonomic stress, suggest a convergent evolutionary theory for having a complete circle of Willis, and potentially clarify reproducibility concerns associated with the FMD test. </p>

<p>Cellular structural mechanics play a key role in homeostasis by transducing mechanical signals to regulate gene expression and by providing adaptive structural stability for the cell. The alteration of nuclear mechanics in various laminopathies and in natural aging can damage these key functions. Arterial endothelial cells appear to be especially vulnerable due to the importance of shear force mechanotransduction to structure and gene regulation as is made evident by the prominent role of atherosclerosis in Hutchinson-Gilford progeria syndrome (HGPS) and in natural aging. Computational models of cellular mechanics may provide a useful tool for exploring the structural hypothesis of laminopathy at the intracellular level. This thesis explores this topic by introducing the biological background of cellular mechanics and lamin proteins in arterial endothelial cells, investigating disease states related to aberrant lamin proteins, and exploring computational models of the cell structure. It then presents a finite element model designed specifically for investigation of nuclear shear forces in arterial endothelial cells. Model results demonstrate that changes in nuclear material properties consistent with those observed in progerin-expressing cells may result in substantial increases in stress concentrations on the nuclear membrane. This supports the hypothesis that progerin disrupts homeostatic regulation of gene expression in response to hemodynamic shear by altering the mechanical properties of the nucleus.</p>

<p dir="ltr">The actin cytoskeleton is a dynamic structure that regulates various important cellular processes, such as cell protrusion, migration, transport, and cell shape changes. Cells employ different actin architectures best suited for each of these functions. We have employed an agent-based model to illuminate how the actin cytoskeleton plays such functions in animal and plant cells, via dynamic interactions between molecular players.</p><p dir="ltr">Lamellipodia found in animal cells are two-dimensional actin protrusion formed on the leading edge of cells, playing an important role in sensing surrounding mechanical environments via focal adhesions. Various molecular players, architecture, and dynamics of the lamellipodia have been investigated extensively during recent decades. Nevertheless, it still remains elusive how each component in the lamellipodia mechanically interacts with each other to attain a stable, dynamic steady state characterized by a retrograde flow emerging in the branched actin network. Using the agent-based model, we investigated how the balance between different subcellular processes is achieved for the dynamic steady state. We simulated a branched network found in the lamellipodia, consisting of actin filament (F-actin), myosin motor, Arp2/3 complex, and actin crosslinking protein. We found the importance of a balance between F-actin assembly at the leading edge of cells and F-actin disassembly at the rear end of the lamellipodia. We also found that F-actin severing is crucial to allow for the proper disassembly of an actin bundle formed via network contraction induced by motor activity. In addition, it was found that various dynamic steady states can exist.</p><p dir="ltr">The actin cytoskeleton in plant cells plays a crucial role in intracellular transport and cytoplasmic streaming, and its structure is very different from the actin cytoskeleton in animal cells. The plant actin cytoskeleton is known to show distinct dynamic behaviors with homeostasis. We used the agent-based model to simulate the plant actin cytoskeleton with the consideration of the key governing mechanisms, including F-actin polymerization/depolymerization, different types of F-actin nucleation events, severing, and capping. We succeeded in reproducing experimental observations in terms of F-actin density, length, nucleation frequency, and rates of severing, polymerization, and depolymerization. We found that the removal of nucleators results in lower F-actin density in the network, which supports recent experimental findings.</p>