Effect of tubular inhomogeneities on feedback-mediated dynamics of a model of a thick ascending limb

Optimizing SGLT inhibitor treatment for diabetes with chronic kidney diseases

Diabetes induces glomerular hyperfiltration, affects kidney function, and may lead to chronic kidney diseases. A novel therapeutic treatment for diabetic patients targets the sodium-glucose cotransporter isoform 2 (SGLT2) in the kidney. SGLT2 inhibitors enhance urinary glucose, [Formula: see text] and fluid excretion and lower hyperglycemia in diabetes by inhibiting [Formula: see text] and glucose reabsorption along the proximal convoluted tubule. A goal of this study is to predict the effects of SGLT2 inhibitors in diabetic patients with and without chronic kidney diseases. To that end, we applied computational rat kidney models to assess how SGLT2 inhibition affects renal solute transport and metabolism when nephron population are normal or reduced (the latter simulates chronic kidney disease). The model predicts that SGLT2 inhibition induces glucosuria and natriuresis, with those effects enhanced in a remnant kidney. The model also predicts that the [Formula: see text] transport load and thus oxygen consumption of the S3 segment are increased under SGLT2 inhibition, a consequence that may increase the risk of hypoxia for that segment. To protect the vulnerable S3 segment, we explore dual SGLT2/SGLT1 inhibition and seek to determine the optimal combination that would yield sufficient urinary glucose excretion while limiting the metabolic load on the S3 segment. The model predicts that the optimal combination of SGLT2/SGLT1 inhibition lowers the oxygen requirements of key tubular segments, but decreases urine flow and [Formula: see text] excretion; the latter effect may limit the cardiovascular protection of the treatment.

A computational model of epithelial solute and water transport along a human nephron

We have developed the first computational model of solute and water transport from Bowman space to the papillary tip of the nephron of a human kidney. The nephron is represented as a tubule lined by a layer of epithelial cells, with apical and basolateral transporters that vary according to cell type. The model is formulated for steady state, and consists of a large system of coupled ordinary differential equations and algebraic equations. Model solution describes luminal fluid flow, hydrostatic pressure, luminal fluid solute concentrations, cytosolic solute concentrations, epithelial membrane potential, and transcellular and paracellular fluxes. We found that if we assume that the transporter density and permeabilities are taken to be the same between the human and rat nephrons (with the exception of a glucose transporter along the proximal tubule and the H⁺-pump along the collecting duct), the model yields segmental deliveries and urinary excretion of volume and key solutes that are consistent with human data. The model predicted that the human nephron exhibits glomerulotubular balance, such that proximal tubular Na⁺ reabsorption varies proportionally to the single-nephron glomerular filtration rate. To simulate the action of a novel diabetic treatment, we inhibited the Na⁺-glucose cotransporter 2 (SGLT2) along the proximal convoluted tubule. Simulation results predicted that the segment’s Na⁺ reabsorption decreased significantly, resulting in natriuresis and osmotic diuresis.

In addition to its well-known function of waste removal from the body, the kidney is also responsible for the critical regulation of the body’s salt, potassium, acid content, and blood pressure. The kidneys perform these life-sustaining task by filtering and returning to blood stream about 200 quarts of blood every 24 hours. What isn’t returned to blood stream is excreted as urine. The production of urine involves highly complex steps of secretion and reabsorption. To study these processes without employing invasive experimental procedures, we developed the first computational model of the human nephron (which is the functional unit of a kidney). The model contains detailed representation of the transport processes that take place in the epithelial cells that form the walls of the nephron. Using that model, we conducted simulations to predict how much filtered solutes and and water is transported along each individual and functionally distinct nephron segment. We conducted these simulations under normal physiological conditions, and under pharmacological conditions. The nephron model can be used as an essential component in an integrated model of kidney function in humans.

A Multicellular Vascular Model of the Renal Myogenic Response

The myogenic response is a key autoregulatory mechanism in the mammalian kidney. Triggered by blood pressure perturbations, it is well established that the myogenic response is initiated in the renal afferent arteriole and mediated by alterations in muscle tone and vascular diameter that counterbalance hemodynamic perturbations. The entire process involves several subcellular, cellular, and vascular mechanisms whose interactions remain poorly understood. Here, we model and investigate the myogenic response of a multicellular segment of an afferent arteriole. Extending existing work, we focus on providing an accurate—but still computationally tractable—representation of the coupling among the involved levels. For individual muscle cells, we include detailed Ca2+ signaling, transmembrane transport of ions, kinetics of myosin light chain phosphorylation, and contraction mechanics. Intercellular interactions are mediated by gap junctions between muscle or endothelial cells. Additional interactions are mediated by hemodynamics. Simulations of time-independent pressure changes reveal regular vasoresponses throughout the model segment and stabilization of a physiological range of blood pressures (80–180 mmHg) in agreement with other modeling and experimental studies that assess steady autoregulation. Simulations of time-dependent perturbations reveal irregular vasoresponses and complex dynamics that may contribute to the complexity of dynamic autoregulation observed in vivo. The ability of the developed model to represent the myogenic response in a multiscale and realistic fashion, under feasible computational load, suggests that it can be incorporated as a key component into larger models of integrated renal hemodynamic regulation.

Theoretical assessment of the Ca2+ oscillations in the afferent arteriole smooth muscle cell of the rat kidney

The afferent arteriole (AA) of rat kidney exhibits the myogenic response, in which the vessel constricts in response to an elevation in blood pressure and dilates in response to a pressure reduction. Additionally, the AA exhibits spontaneous oscillations in vascular tone at physiological luminal pressures. These time-periodic oscillations stem from the dynamic exchange of Ca[Formula: see text] between the cytosol and the sarcoplasmic reticulum, coupled to the stimulation of Ca[Formula: see text]-activated potassium and chloride channels, and to the modulation of voltage-gated L-type Ca[Formula: see text] channels. The effects of physiological factors, including blood pressure and vasoactive substances, on AA vasomotion remain to be well characterized. In this paper, we analyze a mathematical model of Ca[Formula: see text] signaling in an AA smooth muscle cell. The model represents detailed transmembrane ionic transport, intracellular Ca[Formula: see text] dynamics as well as kinetics of nitric oxide (NO) and superoxide (O[Formula: see text]) formation, diffusion and reaction. NO is an important factor in the maintenance of blood pressure and O[Formula: see text] has been shown to contribute significantly to the functional alternations of blood vessels in hypertension. We perform a bifurcation analysis of the model equations to assess the effect of luminal pressure, NO and O[Formula: see text] on the behaviors of limit cycle oscillations.

Modeling the effects of positive and negative feedback in kidney blood flow control

Blood flow in the mammalian kidney is tightly autoregulated. One of the important autoregulation mechanisms is the myogenic response, which is activated by perturbations in blood pressure along the afferent arteriole. Another is the tubuloglomerular feedback, which is a negative feedback that responds to variations in tubular fluid [Cl(-)] at the macula densa.(1) When initiated, both the myogenic response and the tubuloglomerular feedback adjust the afferent arteriole muscle tone. A third mechanism is the connecting tubule glomerular feedback, which is a positive feedback mechanism located at the connecting tubule, downstream of the macula densa. The connecting tubule glomerular feedback is much less well studied. The goal of this study is to investigate the interactions among these feedback mechanisms and to better understand the effects of their interactions. To that end, we have developed a mathematical model of solute transport and blood flow control in the rat kidney. The model represents the myogenic response, tubuloglomerular feedback, and connecting tubule glomerular feedback. By conducting a bifurcation analysis, we studied the stability of the system under a range of physiologically-relevant parameters. The bifurcation results were confirmed by means of a comparison with numerical simulations. Additionally, we conducted numerical simulations to test the hypothesis that the interactions between the tubuloglomerular feedback and the connecting tubule glomerular feedback may give rise to a yet-to-be-explained low-frequency oscillation that has been observed in experimental records.

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.

Mathematical modeling of urea transport in the kidney

Mathematical modeling techniques have been useful in providing insights into biological systems, including the kidney. This article considers some of the mathematical models that concern urea transport in the kidney. Modeling simulations have been conducted to investigate, in the context of urea cycling and urine concentration, the effects of hypothetical active urea secretion into pars recta. Simulation results suggest that active urea secretion induces a "urea-selective" improvement in urine concentrating ability. Mathematical models have also been built to study the implications of the highly structured organization of tubules and vessels in the renal medulla on urea sequestration and cycling. The goal of this article is to show how physiological problems can be formulated and studied mathematically, and how such models may provide insights into renal functions.

Mathematical modeling of renal hemodynamics in physiology and pathophysiology

In addition to the excretion of metabolic waste and toxin, the kidney plays an indispensable role in regulating the balance of water, electrolyte, acid-base, and blood pressure. For the kidney to maintain proper functions, hemodynamic control is crucial. In this review, we describe representative mathematical models that have been developed to better understand the kidney's autoregulatory processes. We consider mathematical models that simulate glomerular filtration, and renal blood flow regulation by means of the myogenic response and tubuloglomerular feedback. We discuss the extent to which these modeling efforts have expanded the understanding of renal functions in health and disease.

Mathematical modeling of kidney transport

In addition to metabolic waste and toxin excretion, the kidney also plays an indispensable role in regulating the balance of water, electrolytes, nitrogen, and acid-base. In this review, we describe representative mathematical models that have been developed to better understand kidney physiology and pathophysiology, including the regulation of glomerular filtration, the regulation of renal blood flow by means of the tubuloglomerular feedback mechanisms and of the myogenic mechanism, the urine concentrating mechanism, epithelial transport, and regulation of renal oxygen transport. We discuss the extent to which these modeling efforts have expanded our understanding of renal function in both health and disease.

Tubular fluid flow and distal NaCl delivery mediated by tubuloglomerular feedback in the rat kidney

The glomerular filtration rate in the kidney is controlled, in part, by the tubuloglomerular feedback (TGF) system, which is a negative feedback loop that mediates oscillations in tubular fluid flow and in fluid NaCl concentration of the loop of Henle. In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a short loop of Henle with compliant walls. The proximal tubule and the outer-stripe segment of the descending limb are assumed to be highly water permeable; the thick ascending limb (TAL) is assumed to be water impermeable and have active NaCl transport. A bifurcation analysis of the TGF model equations was performed by computing parameter boundaries, as functions of TGF gain and delay, that separate differing model behaviors. The analysis revealed a complex parameter region that allows a variety of qualitatively different model equations: a regime having one stable, time-independent steady-state solution and regimes having stable oscillatory solutions of different frequencies. A comparison with a previous model, which represents only the TAL explicitly and other segments using phenomenological relations, indicates that explicit representation of the proximal tubule and descending limb of the loop of Henle lowers the stability of the TGF system. Model simulations also suggest that the onset of limit-cycle oscillations results in increases in the time-averaged distal NaCl delivery, whereas distal fluid delivery is not much affected.

Modeling Transport and Flow Regulatory Mechanisms of the Kidney

The kidney plays an indispensable role in the regulation of whole-organism water balance, electrolyte balance, and acid-base balance, and in the excretion of metabolic wastes and toxins. In this paper, we review representative mathematical models that have been developed to better understand kidney physiology and pathophysiology, including the regulation of glomerular filtration, the regulation of renal blood flow by means of the tubuloglomerular feedback mechanisms and of the myogenic mechanism, the urine concentrating mechanism, and regulation of renal oxygen transport. We discuss how such modeling efforts have significantly expanded our understanding of renal function in both health and disease.