Analysis of Models of Coupled Nephrons

Dissertation Research. My dissertation research centers on the investigation of a mathematical model of the feedback system that regulates the rate of filtration in nephrons. The nephron is the functional unit of the kidney; in the nephron, tubular fluid (nearly identical to blood plasma) is filtered out of the bloodstream and sodium chloride levels in the blood are maintained by the reabsorption of water and solutes from the nephron back into the blood. The blood flows from the afferent arteriole into a capillary bundle, the glomerulus, from which the filtrate enters the proximal tubule. From there the tubular fluid flows into the loop of Henle (consisting of the descending limb and thick ascending limb) from which sodium chloride, other solutes, and water are absorbed into the local vasculature. What remains flows into the distal convoluted tubule and collecting duct to be excreted as urine. Figure 1 depicts a schematic of two (coupled) nephrons fed from the same afferent arteriole. My research involves tubuloglomerular feedback (TGF), the key moment-to-moment controller of the rate of filtration of tubular fluid through the nephron. TGF plays an important role in regulating the transit time of fluid through the nephron that in turn determines the amount of sodium chloride which is reabsorbed into the blood. I am studying the effects of the coupled nephrons via TGF signals. Working in the context of a previously developed model of this process [5] (a semi-linear hyperbolic PDE with a time delay), I have developed an efficient numerical algorithm for computing numerical solutions to the model and used this numerical simulation to investigate and understand the behavior of the model in several different parameter regimes. Also, via numerical calculation of solutions to the model’s characteristic equation, I have located the boundaries of the regions in which these different behaviors occur. I have substantially increased understanding of the more complex behaviors through analysis of discrete Fourier transforms of time series generated by numerical simulation. Also, I am a co-author of a manuscript [7] that treats the effects of coupling. Although the manuscript was in a fairly mature draft when I began work on the project, I was able to assist by checking calculations, deepening understanding of results, and correcting several errors. My current research has suggested several problems that I intend to work on in the near future. One problem involves a simplified model of coupled feedback oscillators that exhibits