Browsing by Author "Layton, Anita T"

The main functions of the kidney take place in the nephrons. For their proper operation, nephrons need to be supplied with a stable blood flow that remains constant despite fluctuations of arterial pressure. Such stability is provided by the afferent arterioles, which are unique vessels in the kidney capable of adjusting diameter. By doing so, afferent arterioles regulate blood delivery downstream, where the nephrons are located. The afferent arterioles respond to signals initiated by two mechanisms: the myogenic response which operates to absorb pressure perturbations within the vasculature, and tubuloglomerular feedback which operates to stabilize salt reabsorption.

In this thesis, a mathematical model of the renal nephrovasculature that represents both mechanisms in a dynamical context is developed. For this purpose, de- tailed representations of the myogenic mechanism of vascular smooth muscles and the tubular processes are developed and combined in a single comprehensive model. The resulting model is formulated with a large number of ordinary differential equations that represent the intracellular processes of arteriolar smooth muscles, coupled with a number of partial differential equations, mainly of the advection-diffusion-reaction type, that represent blood flow, glomerular filtration and the tubular processes. Due to its unique activation characteristics, the myogenic response is formulated with a set of delay differential equations.

The model is utilized to assess a verity of physiological phenomena: the conduction of vasomotor responses along the afferent arteriole, autoregulation under physiologic as well as pathophysiologic conditions, and renal oxygenation. A first attempt to model the impact of diabetes mellitus on renal hemodynamics is also made. Further, an application with clinical significance is presented. Namely, renal oxygenation is estimated under conditions that simulate those observed during cardiopulmonary surgery. Results indicate the development of renal hypoxia, which suggests an important pathway for the development of acute kidney injury.

Blood flow can be modeled as a fluid-structure interaction problem in which the vessel is represented as an infinitely thin elastic interface that exerts a singular force on the internal and surrounding fluid. The immersed interface method was created to solve this type of immersed boundary problem with second-order accuracy in space and time. However, the interface must be a closed shape, which is not conducive to modeling flow in a vessel.

An extension of the immersed interface method to also solve immersed boundary problems where the interface is shaped like an open tube that transverses the fluid domain is presented. Numerical results indicate that this method converges with second order in both space and time and can sharply capture discontinuities in the fluid solutions.

Additionally, mathematical models for simulating renal blood flow under physiological and pathophysiological conditions are presented.

In particular, models simulating the myogenic response to changes in systolic blood pressure in the afferent arteriole and models simulating

the effect of pericyte contractions on vascular congestion in the descending vasa recta is considered.

One of the key mechanisms that mediate renal autoregulation is the tubuloglomerular feedback (TGF) system, which is a negative feedback loop in the kidney that balances glomerular filtration with tubular reabsorptive capacity. In this dissertation, we develop several mathematical models of the TGF system to study TGF-mediated model dynamics.

First, we develop a mathematical model of compliant thick ascending limb (TAL) of a short loop of Henle in the rat kidney, called TAL model, to investigate the effects of spatial inhomogeneous properties in TAL on TGF-mediated dynamics. We derive a characteristic equation that corresponds to a linearized TAL model, and conduct a bifurcation analysis by finding roots of that equation. Results of the bifurcation analysis are also validated via numerical simulations of the full model equations.

We then extend the TAL model to explicitly represent an entire short-looped nephron including the descending segments and having compliant tubular walls, developing a short-looped nephron model. A bifurcation analysis for the TGF loop-model equations is similarly performed by computing parameter boundaries, as functions of TGF gain and delay, that separate differing model behaviors. We also use the loop model to better understand the effects of transient as well as sustained flow perturbations on the TGF system and on distal NaCl delivery.

To understand the impacts of internephron coupling on TGF dynamics, we further develop a mathematical model of a coupled-TGF system that includes any finite number of nephrons coupled through their TGF systems, coupled-nephron model. Each model nephron represents a short loop of Henle having compliant tubular walls, based on the short-looped nephron model, and is assumed to interact with nearby nephrons through electrotonic signaling along the pre-glomerular vasculature. The characteristic equation is obtained via linearization of the loop-model equations as in TAL model. To better understand the impacts of parameter variability on TGF-mediated dynamics, we consider special cases where the relation between TGF delays and gains among two coupled nephrons is specifically chosen. By solving the characteristic equation, we determine parameter regions that correspond to qualitatively differing model behaviors.

TGF delays play an essential role in determining qualitatively and quantitatively different TGF-mediated dynamic behaviors. In particular, when noise arising from external sources of system is introduced, the dynamics may become significantly rich and complex, revealing a variety of model behaviors owing to the interaction with delays. In our next study, we consider the effect of the interactions between time delays and noise, by developing a stochastic model. We begin with a simple time-delayed transport equation to represent the dynamics of chloride concentration in the rigid-TAL fluid. Guided by a proof for the existence and uniqueness of the steady-state solution to the deterministic Dirichlet problem, obtained via bifurcation analysis and the contraction mapping theorem, an analogous proof for stochastic system with random boundary conditions is presented. Finally we conduct multiscale analysis to study the effect of the noise, specifically when the system is in subcritical region, but close enough to the critical delay. To analyze the solution behaviors in long time scales, reduced equations for the amplitude of solutions are derived using multiscale method.

This dissertation introduces the new computational methods for two major topics. The first topic is computing the Stokes flow driven by an open immersed interface. The other topic is the simulation of the Stokes and Navier-Stokes fluid through an elastic tube driven by an internal source and sink.

For the first topic, we developed two second-order accurate method. One is for accurately evaluating boundary integral solutions at a point, and the other is for computing Stokes solution values on a rectangular mesh. We first describe a method for computing singular or nearly singular integrals, evaluated at a point on or near the curve. To improve accuracy of the numerical quadrature, we add corrections for the errors arising from discretization, which are found by asymptotic analysis. When used to solve the Stokes equations with sources on an open, immersed interface, the method generates second-order approximations, for both the pressure and the velocity, and preserves the jumps in the solutions and their derivatives across the boundary. We then combine the method with a mesh-based solver to yield a hybrid method for computing Stokes solutions at N2 grid points on a rectangular grid. Numerical results are presented which exhibit second-order accuracy. To demonstrate the applicability of the method, we use the method to simulate fluid dynamics induced by the beating motion of a cilium.

For the second topic, we present numerical method for simulating both Stokes and Navier Stokes fluid flow through a compliant, closed tube, driven by an internal source and sink. The governing equations are implemented in axisymmetric cylindrical coordinates, which capture 3D flow dynamics with only 2D computations.

In the Stokes fluid flow simulations, we solve the model equations using a hybrid approach: we decompose the pressure and velocity fields into parts due to the surface force and due to the source and sink, with each part handled separately by means of an appropriate method. Because the singularly-supported surface force yields an unsmooth solution, that part of the solution is computed by using the immersed interface method with the jump conditions for the axisymmetric cylindrical coordinates. The velocity due to the source and sink is calculated along the tubular surface using boundary integrals. The source and sink are prescribed in the simulation. From the convergence test and oscillating frequency-amplitude study, we can demonstrate second-order accuracy and applicability of the method.

In the Navier-Stokes flow simulations, we adopt the velocity decomposition approach developed by Beale and Layton. The total velocity is decomposed into the Stokes part and the regular part. The Stokes part satisfies the Stokes equation and includes the boundary force. The regular part satisfies the modified Navier-Stokes equation that incorporate the source and sink terms, with the latter computed using the Hagen-Poiseuille equation. Convergence test, oscillating frequency-amplitude study and fluid viscosity-amplitude study are presented that demonstrate the accuracy of the method.

Hypertension is a global health challenge: it affects one billion people worldwide and is estimated to account for >60% of all cases or types of cardiovascular disease. Due to our partial understanding of sex differences in blood pressure regulation mechanisms, fewer hypertensive women achieve blood pressure control compared to men, even though compliance and treatment rates are generally higher in women. Furthermore, concurrent use of typical antihypertensive treatments such as a diuretic, a renin-angiotensin system (RAS) inhibitor, and a non-steroidal anti-inflammatory drug (NSAID) significantly increases the risk of acute kidney injury (AKI). This phenomenon is known as “triple whammy” AKI. Diuretics and RAS inhibitors are often prescribed in tandem for the treatment of hypertension, whereas some NSAIDs, such as ibuprofen, are available over the counter. As such, concurrent treatment with all three drugs is common.

Thus, the objective of this study is to identify which factors contribute to the sexual dimorphism in response to anti-hypertensive therapies targeting the RAS. We also aim to better understand the mechanisms underlying the development of triple whammy AKI and to identify physiological factors that may increase an individual’s susceptibility.

To accomplish these goals, we develop sex-specifc models of blood pressure regulation in humans. Model components include variables describing the heart and circulation, kidney function, sodium and water reabsorption in the nephron, and the RAS. Sex differences in the RAS, baseline aldosterone level, and the reactivity of renal sympathetic nervous activity (RSNA) are represented.

Model results suggest that the main source of sexual dimorphism in treatment efficacy is how the effects of the bound RAS receptors differ between males and females -- specifically the feedback mechanisms of the angiotensin II type 1 receptor on renin secretion and the effects of the angiotensin II type two receptor on renal resistance. In regards to triple whammy AKI, model simulations suggest that individual variations in water intake or the myogenic response as well as high dosages of these drugs may predispose patients with hypertension to develop triple whammy AKI.

These proposed models hold great potential for extensions to study other components of blood pressure regulation, such as the interconnectedness of K+ regulation and Na+ regulation. We present a model of K+ regulation including the aldosterone and renal function feedback controls, as well as the feedforward control stimulated by dietary K+ intake. Model results suggest that the feedforward effect is necessary for increased urinary K+ excretion during digestion and that muscle-kidney cross talk can accelerate recovery following perturbations in extracellular K+ concentration.